number-theory

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April Fools Day Contest

April Fools Day Contest Type: April Fools | Problems: 8 | Verified: 8/8 | Rating range: 1200-2000 | Time: 13m 15s Problem Name Rating Tags Solve Time Verified A Mysterious numbers - 1 1200 *special, constructive-algorithms 1m 43s ✓ B Star 1300 *special, combinatorics 2m 57s ✓ C A Piece of Cake 2000 *special, implementation 1m 30s ✓ D Broken checker 1300 *special, brute-force 1m 14s ✓ E MYSTERIOUS LANGUAGE...

Practice › Codeforces Solutions › April Fools Day Contest ›

CF 171F - ucyhf

CF 171F - ucyhf Rating: 1600 Tags: *special, brute force, implementation, number theory Solve time: 1m 11s Verified: yes Solution Problem Understanding We are asked to find a special number associated with a single integer input d , where d represents a divisor or parameter in a number-theoretic sequence. The output is another integer computed from d according to a hidden mathematical rule, likely involving divisibility, modular arithmetic, or a...

Practice › Codeforces Solutions › Codeforces Round 1090 (Div. 4) ›

CF 2218D - The 67th OEIS Problem

CF 2218D - The 67th OEIS Problem Rating: 1100 Tags: constructive algorithms, greedy, math, number theory Solve time: 2m 4s Verified: no Solution Problem Understanding We are asked to construct sequences of integers of length $n$ such that the greatest common divisor (gcd) of consecutive elements is always distinct. Each test case provides a number $n$, and for each, we must output one valid sequence of length $n$. The output...

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Codeforces Round 1093 (Div. 1)

Codeforces Round 1093 (Div. 1) Type: Div. 1 | Problems: 1 | Verified: 0/1 | Rating range: - | Time: 3m 30s Problem Name Rating Tags Solve Time Verified A Grid L - brute-force, constructive-algorithms, math 3m 30s ✗

Practice › Codeforces Solutions › Codeforces Round 1093 (Div. 1) ›

CF 2219A - Grid L

CF 2219A - Grid L Rating: - Tags: brute force, constructive algorithms, math, number theory Solve time: 3m 30s Verified: no Solution Problem Understanding An $n \times m$ grid consists of all unit segments that form the lattice. Every cell contributes edges, but neighboring cells share edges, so the total number of unit segments in the whole grid is not $4nm$. We are given two types of pieces. A straight...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.9 Factorization

18.9 Factorization Many number-theoretic algorithms become simpler once a number is expressed as a product of primes. Divisor counting, divisor sums, Euler's phi function, modular arithmetic, primality testing, cryptography, and multiplicative functions all rely on prime factorization. The Fundamental Theorem of Arithmetic states that every integer greater than one can be represented uniquely as a product of prime powers. Factorization algorithms exploit this fact by repeatedly discovering prime divisors and...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.22 Practical Number Theory Patterns

18.22 Practical Number Theory Patterns Throughout this chapter we studied individual algorithms: Euclidean Algorithm Modular Arithmetic Fast Exponentiation Sieve Methods Chinese Remainder Theorem Euler Phi Primitive Roots Miller-Rabin Pollard's Rho Möbius Inversion In real systems, however, these algorithms rarely appear in isolation. Practical number theory is about recognizing recurring patterns and selecting the right combination of tools. This section serves as a cookbook of common problem types, showing how experienced...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.14 Möbius Function and Möbius Inversion

18.14 Möbius Function and Möbius Inversion Many counting problems involve inclusion-exclusion. We count objects satisfying one condition, subtract those satisfying another, add back intersections, and continue alternating signs. The Möbius function packages this process into a powerful algebraic tool. At first glance, the Möbius function appears obscure. In practice, it provides elegant solutions to divisor-sum problems, coprime counting, multiplicative-function analysis, and inversion of arithmetic relationships. Once understood, it becomes one...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.23 Complexity Analysis

18.23 Complexity Analysis Number theory algorithms often look fast because their code is short. That can be misleading. A loop that runs log n times may still be expensive if each iteration multiplies thousand-bit integers. A primality test may look constant-time for machine integers but behave very differently for arbitrary-precision inputs. This section collects the complexity models used throughout the chapter and shows how to reason about number-theoretic algorithms in...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.2 Euclidean Algorithm

18.2 Euclidean Algorithm The Euclidean Algorithm is one of the oldest known algorithms and remains one of the most useful. It computes the greatest common divisor (GCD) of two integers efficiently, even when the numbers contain hundreds or thousands of digits. Many advanced algorithms depend directly on GCD computation. Modular inverses, rational arithmetic, cryptographic systems, fraction simplification, number-theoretic transforms, and factorization methods all rely on this fundamental procedure. The remarkable...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.4 Modular Arithmetic

18.4 Modular Arithmetic Modular arithmetic is the arithmetic of remainders. Instead of tracking the exact value of an integer, we track only its position within a finite cycle. This simple idea transforms many seemingly impossible computations into efficient algorithms and forms the foundation of modern cryptography, hashing, number theory, combinatorics, distributed systems, and competitive programming. A computer clock provides an intuitive example. If the time is 10 o'clock and we...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.10 Chinese Remainder Theorem

18.10 Chinese Remainder Theorem The Chinese Remainder Theorem is a reconstruction tool. It lets you replace one computation modulo a large number with several computations modulo smaller numbers, then combine the answers into a unique result modulo the product of those smaller moduli. This pattern appears in number theory, cryptography, high-performance arithmetic, coding theory, calendar calculations, and algorithmic problem solving. It is especially useful when the smaller moduli are pairwise...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.5 Modular Inverse

18.5 Modular Inverse Division is one of the most subtle operations in modular arithmetic. Addition, subtraction, and multiplication always produce valid results under a modulus, but division only works under specific conditions. Understanding these conditions is essential because modular division appears throughout cryptography, combinatorics, polynomial arithmetic, number-theoretic transforms, and countless algorithmic problems. The key idea is that modular arithmetic has no direct division operator. Instead, division is replaced by multiplication...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.13 Modular Combinatorics

18.13 Modular Combinatorics Many counting problems ask for results modulo a large number. The answer may be enormous, but the required output is usually: answer mod M This is common in combinatorics, dynamic programming, graph counting, probability calculations, and programming-contest problems. Modular combinatorics provides the tools for computing factorials, permutations, combinations, and related counting formulas without ever constructing the full integer answer. The most common setting uses a large prime...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.20 Pollard's Rho Factorization

18.20 Pollard's Rho Factorization Primality testing tells us whether a number is prime. Factorization asks a harder question: If a number is composite, what are its prime factors? For small integers, trial division is sufficient. For large integers, trial division becomes impractical because it requires checking divisors up to approximately: √n Pollard's Rho is one of the most important practical factorization algorithms. It is simple to implement, surprisingly effective, and...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.1 Divisibility

18.1 Divisibility Divisibility is one of the most fundamental concepts in number theory. Nearly every algorithm in computational number theory relies on divisibility properties, whether directly or indirectly. Prime factorization, greatest common divisors, modular arithmetic, congruences, cryptographic protocols, and arithmetic functions all begin with the simple question: Does one integer divide another? Despite its apparent simplicity, divisibility provides a rich algebraic structure that allows many difficult problems to be transformed...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.6 Fast Exponentiation

18.6 Fast Exponentiation Exponentiation appears constantly in number-theoretic algorithms. Modular arithmetic, primality testing, cryptography, polynomial hashing, matrix recurrence solving, and combinatorial counting all depend on computing powers efficiently. The direct method is simple: multiply the base by itself repeatedly. That works for small exponents, but fails quickly when the exponent is large. Fast exponentiation, also called exponentiation by squaring, reduces the number of multiplications from linear to logarithmic. Problem Given:...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.25 Number Theory Toolkit

18.25 Number Theory Toolkit After working through the algorithms in this chapter, it is useful to step back and assemble them into a practical toolkit. Real-world problems rarely announce which technique they require. Instead, they present constraints, equations, divisibility conditions, counting requirements, or modular arithmetic expressions. Your task is to recognize the pattern and select the appropriate tool. This final section summarizes the essential algorithms, implementation templates, decision strategies, and...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.7 Primality Testing

18.7 Primality Testing Prime numbers occupy a central position in number theory. They are the building blocks of the integers, appear throughout algebraic structures, and form the foundation of modern public-key cryptography. Many algorithms depend on quickly determining whether a given integer is prime. At first glance, primality testing seems simple. A number is prime if it has no divisors other than 1 and itself. However, when the input contains...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.17 Primitive Roots

18.17 Primitive Roots Primitive roots describe generators of modular multiplication. Instead of looking at all nonzero residues modulo a prime as unrelated values, a primitive root lets you express every nonzero residue as a power of one number. This idea is central to discrete logarithms, finite fields, cryptographic protocols, cyclic groups, number-theoretic transforms, and modular algebra. In algorithmic number theory, primitive roots are useful whenever a multiplicative structure needs to...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.19 Miller-Rabin Primality Test

18.19 Miller-Rabin Primality Test The Miller-Rabin test is the practical standard for fast primality testing. It improves on Fermat's test by checking not only whether a number satisfies a modular exponent identity, but also whether its square-root structure behaves like a prime. For large integers, trial division is too slow. Fermat testing is faster, but can be fooled by Carmichael numbers. Miller-Rabin keeps the speed of modular exponentiation while detecting...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.18 Discrete Logarithms

18.18 Discrete Logarithms Ordinary logarithms answer the question: a^x = b What is x ? Discrete logarithms ask the same question in modular arithmetic: g^x ≡ h (mod p) Given: g h p find: x Unlike ordinary logarithms, there is no simple closed-form formula. For sufficiently large primes, finding x is computationally difficult. This difficulty forms the foundation of many public-key cryptographic systems. From an algorithmic perspective, discrete logarithms are...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.12 Euler's Theorem and Fermat's Little Theorem

18.12 Euler's Theorem and Fermat's Little Theorem Many algorithms require computing enormous powers modulo an integer. At first glance, expressions such as: 7^1000000 mod 13 appear computationally impossible. However, modular arithmetic contains deep structural properties that allow such expressions to be simplified dramatically. Two of the most important results are Fermat's Little Theorem and Euler's Theorem. These theorems explain how powers behave in modular systems and provide the theoretical foundation...

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Chapter 18. Number Theory

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.21 Big Integer Concerns

18.21 Big Integer Concerns Number-theoretic algorithms often look simple when written with mathematical notation. In code, the details are less forgiving. Modular multiplication can overflow. Exponentiation can allocate huge temporary values. Division may dominate runtime. Randomized algorithms may behave differently across platforms. Cryptographic code may leak information through timing. Big integer concerns are the implementation layer beneath computational number theory. They determine whether an otherwise correct algorithm remains correct, fast,...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.11 Euler Phi Function

18.11 Euler Phi Function Euler's phi function counts how many numbers in a range are coprime to a given integer. It appears naturally in modular arithmetic, cryptography, primitive roots, multiplicative functions, and counting problems over residue classes. The function is usually written as: φ(n) It counts the positive integers k such that: 1 ≤ k ≤ n and: gcd(k,n) = 1 For example: φ(9) = 6 because the numbers coprime...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.3 Extended Euclidean Algorithm

18.3 Extended Euclidean Algorithm The Euclidean Algorithm computes the greatest common divisor of two integers. In many applications, however, the GCD alone is not enough. We often need to know how the GCD can be expressed as a linear combination of the original numbers. Given integers a and b , we want to find integers x and y such that: ax + by = gcd(a,b) This problem appears throughout number...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.24 Testing Number-Theoretic Code

18.24 Testing Number-Theoretic Code Number-theoretic code is compact, but its edge cases are dense. A function may work for most random inputs and still fail on zero, one, negative values, non-coprime moduli, repeated prime factors, overflow boundaries, or composite numbers that imitate primes. Testing these algorithms requires more than checking a few examples. Good tests combine known values, algebraic identities, brute-force oracles for small inputs, randomized comparison, and reconstruction checks....

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.15 Linear Congruences

18.15 Linear Congruences A linear congruence is the modular analogue of a linear equation. Instead of solving ax = b over the integers or real numbers, we solve ax ≡ b (mod m) over residue classes. This problem appears in modular division, cyclic scheduling, cryptography, Diophantine equations, calendar arithmetic, and many CRT-based algorithms. The main difference from ordinary algebra is that division is not always valid. The coefficient a may...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.8 Sieve of Eratosthenes

18.8 Sieve of Eratosthenes The Sieve of Eratosthenes is the standard algorithm for generating all primes up to a limit. Instead of testing each number independently, it eliminates composites in bulk. This makes it much faster than repeated trial division when many primes are needed. The sieve is useful whenever an algorithm needs a table of primes, fast primality lookup for bounded values, or precomputed arithmetic information. It appears in...

Programming › Algorithm Cookbook › Chapter 18. Number Theory ›

18.16 Diophantine Equations

18.16 Diophantine Equations A Diophantine equation is an equation whose solutions must be integers. In algorithmic number theory, the most common form is the linear Diophantine equation: ax + by = c where a , b , and c are given integers, and we need integer values of x and y . This problem is closely related to the Extended Euclidean Algorithm, modular inverses, and linear congruences. Once you know...

Practice › Codeforces Solutions › Codeforces Round 1095 (Div. 2) ›

CF 2226B - Everything Everywhere

CF 2226B - Everything Everywhere Rating: - Tags: greedy, math, number theory Solve time: 3m 22s Verified: no Solution Problem Understanding We are given a permutation of the integers from 1 to n . For every contiguous subarray, we look at three quantities: its maximum value, its minimum value, the GCD of all values inside it. A subarray is called good when $$\max - \min = \gcd(\text{all elements})$$ The task...

Practice › Codeforces Solutions › Codeforces Round 1095 (Div. 2) ›

CF 2226F - Inversion Invasion

CF 2226F - Inversion Invasion Rating: - Tags: math, number theory Solve time: 1m 41s Verified: no Solution Problem Understanding For every divisor $d\mid n$, define $$G_d={v\in[1,n]\mid \gcd(v,n)=d}.$$ The array $a$ gradually fixes some positions. If $a_i=d\neq 0$, then position $i$ must contain a value from $G_d$. Unfixed positions may contain any remaining values. A valid permutation exists only if, for every divisor $d$, $$c_d \le |G_d|,$$ where $c_d$ is...

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Codeforces Round 1095 (Div. 2)

Codeforces Round 1095 (Div. 2) Type: Div. 2 | Problems: 7 | Verified: 0/7 | Rating range: - | Time: 23m 23s Problem Name Rating Tags Solve Time Verified A Disturbing Distribution - greedy, math 3m 10s ✗ B Everything Everywhere - greedy, math, number-theory 3m 22s ✗ C Mental Monumental (Easy Version) - binary-search, data-structures, greedy 6m 17s ✗ D Reserved Reversals - constructive-algorithms, dp, greedy 2m 22s ✗...

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LeetCode 3671 - Sum of Beautiful Subsequences

LeetCode Problem 3671 Difficulty: 🔴 Hard Topics: Array, Math, Binary Indexed Tree, Number Theory Solution Problem Understanding Let a subsequence of nums be any sequence multiples. citeturn0search1turn0search2 check whether it is strictly increasing, compute its GCD, and add that GCD to the answer. This is correct because every valid subsequence is examined exactly once. Unfortunately, there are 2^n subsequences. Even for `n = um queries efficiently. citeturn0search1turn0 Force | O(2^n...

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LeetCode 3669 - Balanced K-Factor Decomposition

LeetCode Problem 3669 Difficulty: 🟡 Medium Topics: Math, Backtracking, Number Theory Solution Problem Understanding We are given two integers, n and k . Our goal is to split n into exactly k positive integers whose product is equal to n . Among all valid decompositions, we want the one where the difference between the largest number and the smallest number is as small as possible. Since the maximum difference between...

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LeetCode 3610 - Minimum Number of Primes to Sum to Target

LeetCode Problem 3610 Difficulty: 🟡 Medium Topics: Array, Math, Dynamic Programming, Number Theory Solution Problem Understanding The problem requires us to determine the minimum number of prime numbers from the first m primes whose sum equals n . A multiset is allowed, meaning each prime may be chosen multiple times. The input integers n and m specify the target sum and the number of primes available, respectively. The output is...

Practice › Codeforces Solutions › Codeforces Round 937 (Div. 4) ›

CF 1950E - Nearly Shortest Repeating Substring

CF 1950E - Nearly Shortest Repeating Substring Rating: 1500 Tags: brute force, implementation, number theory, strings Solve time: 1m 22s Verified: yes Solution Problem Understanding We are given a string s of length n . We want to find the smallest possible length L such that there exists a pattern string k of length L , repeated exactly n / L times, producing a string c of length n ,...

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Codeforces Round 943 (Div. 3)

Codeforces Round 943 (Div. 3) Type: Div. 3 | Problems: 2 | Verified: 1/2 | Rating range: 800-800 | Time: 2m 13s Problem Name Rating Tags Solve Time Verified A Maximize? 800 brute-force, math, number-theory 1m 6s ✗ B Prefiquence 800 greedy, two-pointers 1m 7s ✓

Practice › Codeforces Solutions › Educational Codeforces Round 164 (Rated for Div. 2) ›

CF 1954C - Long Multiplication

CF 1954C - Long Multiplication Rating: 1200 Tags: greedy, math, number theory Solve time: 1m 13s Verified: no Solution Problem Understanding We are given two very large integers represented as strings, both having the same number of digits. Every digit is between 1 and 9, so there are no zeros to complicate positional effects or leading zero issues. The only operation allowed is to swap digits at the same position...

Practice › Codeforces Solutions › Codeforces Round 941 (Div. 1) ›

CF 1965B - Missing Subsequence Sum

CF 1965B - Missing Subsequence Sum Rating: 1800 Tags: bitmasks, constructive algorithms, greedy, number theory Solve time: 1m 17s Verified: no Solution Problem Understanding We are asked to construct an array of non-negative integers such that the sums of all subsequences cover every integer from 1 to $n$, except for a single forbidden sum $k$. Each test case gives the upper bound $n$ and the forbidden sum $k$. The output...

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Codeforces Round 935 (Div. 3)

Codeforces Round 935 (Div. 3) Type: Div. 3 | Problems: 8 | Verified: 5/8 | Rating range: 800-2600 | Time: 9m 25s Problem Name Rating Tags Solve Time Verified A Setting up Camp 800 greedy, math 55s ✓ B Fireworks 900 math, number-theory 1m 25s ✓ C Left and Right Houses 1200 brute-force 48s ✓ D Seraphim the Owl 1300 dp, greedy 1m 20s ✓ E Binary Search 1700 binary-search,...

Practice › Codeforces Solutions › Codeforces Round 945 (Div. 2) ›

CF 1973F - Maximum GCD Sum Queries

CF 1973F - Maximum GCD Sum Queries Rating: 3100 Tags: bitmasks, brute force, dp, implementation, number theory Solve time: 5m 16s Verified: no Solution

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Codeforces Round 279 (Div. 2)

Codeforces Round 279 (Div. 2) Type: Div. 2 | Problems: 5 | Verified: 0/5 | Rating range: 1500-2200 | Time: 44m 5s Problem Name Rating Tags Solve Time Verified B Queue 1500 dsu, implementation 11m 48s ✗ C Hacking Cypher 1700 brute-force, math, number-theory 5m 25s ✗ D Chocolate 1900 brute-force, dfs-and-similar, math 5m 18s ✗ E Restoring Increasing Sequence 2000 binary-search, brute-force, greedy 10m 45s ✗ F Treeland Tour...

Practice › Codeforces Solutions › Codeforces Round 937 (Div. 4) ›

CF 1950D - Product of Binary Decimals

CF 1950D - Product of Binary Decimals Rating: 1100 Tags: brute force, dp, implementation, number theory Solve time: 1m 17s Verified: yes Solution Problem Understanding We are given up to 50,000 independent queries. For each query, a number n is provided, where 1 ≤ n ≤ 100000 . A number is called a binary decimal if every digit in its usual decimal representation is either 0 or 1 . Examples...

Practice › Codeforces Solutions › Codeforces Round 942 (Div. 1) ›

CF 1967B2 - Reverse Card (Hard Version)

CF 1967B2 - Reverse Card (Hard Version) Rating: 2200 Tags: brute force, math, number theory Solve time: 1m 30s Verified: no Solution Problem Understanding We are asked to count ordered pairs of integers $(a, b)$ with $1 \le a \le n$ and $1 \le b \le m$ such that $b \cdot \gcd(a, b)$ is divisible by $a+b$. The inputs are multiple test cases, each specifying a pair of limits $n$...

Practice › Codeforces Solutions › Codeforces Round 282 (Div. 2) ›

CF 495B - Modular Equations

CF 495B - Modular Equations Rating: 1600 Tags: math, number theory Solve time: 5m 26s Verified: no Solution

Practice › Codeforces Solutions › Codeforces Round 278 (Div. 1) ›

CF 487C - Prefix Product Sequence

CF 487C - Prefix Product Sequence Rating: 2300 Tags: constructive algorithms, math, number theory Solve time: 5m 22s Verified: no Solution

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Codeforces Round 282 (Div. 2)

Codeforces Round 282 (Div. 2) Type: Div. 2 | Problems: 2 | Verified: 1/2 | Rating range: 1100-1600 | Time: 7m 5s Problem Name Rating Tags Solve Time Verified A Digital Counter 1100 implementation 1m 39s ✓ B Modular Equations 1600 math, number-theory 5m 26s ✗

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Codeforces Round 275 (Div. 2)

Codeforces Round 275 (Div. 2) Type: Div. 2 | Problems: 2 | Verified: 0/2 | Rating range: 1100-1800 | Time: 16m 14s Problem Name Rating Tags Solve Time Verified A Counterexample 1100 brute-force, implementation, math 10m 52s ✗ B Friends and Presents 1800 binary-search, math 5m 22s ✗

Practice › Codeforces Solutions › Codeforces Round 279 (Div. 2) ›

CF 490D - Chocolate

CF 490D - Chocolate Rating: 1900 Tags: brute force, dfs and similar, math, meet-in-the-middle, number theory Solve time: 5m 18s Verified: no Solution

Practice › Codeforces Solutions › Codeforces Round 938 (Div. 3) ›

CF 1955G - GCD on a grid

CF 1955G - GCD on a grid Rating: 1900 Tags: brute force, dfs and similar, dp, implementation, math, number theory Solve time: 1m 6s Verified: yes Solution Problem Understanding We are given a rectangular grid of integers. A move starts at the top-left cell and ends at the bottom-right cell, and at each step we can only go either one cell down or one cell right. Every such move sequence...

Practice › Codeforces Solutions › Codeforces Round 943 (Div. 3) ›

CF 1968A - Maximize?

CF 1968A - Maximize? Rating: 800 Tags: brute force, math, number theory Solve time: 1m 6s Verified: no Solution Problem Understanding We are given a small positive integer $x$. For each such value, we need to choose another integer $y$ strictly smaller than $x$, and we want to maximize the expression $\gcd(x, y) + y$. The interaction between the two terms is important. The gcd term grows when $y$ shares...

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CF 490C - Hacking Cypher

CF 490C - Hacking Cypher Rating: 1700 Tags: brute force, math, number theory, strings Solve time: 5m 25s Verified: no Solution

Practice › Codeforces Solutions › Educational Codeforces Round 164 (Rated for Div. 2) ›

CF 1954E - Chain Reaction

CF 1954E - Chain Reaction Rating: 2200 Tags: binary search, data structures, dsu, greedy, implementation, math, number theory Solve time: 1m 12s Verified: no Solution Problem Understanding We are given a line of monsters, each with some initial health. One operation consists of choosing a single monster as the starting point of a “chain lightning”. That lightning hits the chosen monster and then continues outward to both sides, always staying...

Practice › Codeforces Solutions › Codeforces Round 935 (Div. 3) ›

CF 1945B - Fireworks

CF 1945B - Fireworks Rating: 900 Tags: math, number theory Solve time: 1m 25s Verified: yes Solution Problem Understanding Two firework machines start at time zero and then keep launching fireworks periodically. The first machine fires at times that are multiples of a , and the second fires at multiples of b . Every firework, once launched, stays visible for m + 1 minutes, so a firework launched at time...

Practice › Codeforces Solutions › Codeforces Round 942 (Div. 1) ›

CF 1967B1 - Reverse Card (Easy Version)

CF 1967B1 - Reverse Card (Easy Version) Rating: 1400 Tags: brute force, math, number theory Solve time: 57s Verified: no Solution Problem Understanding We are asked to count the number of ordered pairs $(a, b)$ where $1 \le a \le n$ and $1 \le b \le m$ such that the sum $a+b$ is divisible by $b \cdot \gcd(a,b)$. In other words, if you take the greatest common divisor of $a$...

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Codeforces Round 937 (Div. 4)

Codeforces Round 937 (Div. 4) Type: Div. 4 | Problems: 7 | Verified: 6/7 | Rating range: 800-1900 | Time: 7m 41s Problem Name Rating Tags Solve Time Verified A Stair, Peak, or Neither? 800 implementation 1m 4s ✓ B Upscaling 800 implementation 45s ✓ C Clock Conversion 800 implementation, math 51s ✓ D Product of Binary Decimals 1100 brute-force, dp, implementation 1m 17s ✓ E Nearly Shortest Repeating Substring...

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Codeforces Round 278 (Div. 1)

Codeforces Round 278 (Div. 1) Type: Div. 1 | Problems: 4 | Verified: 0/4 | Rating range: 1800-3200 | Time: 27m 2s Problem Name Rating Tags Solve Time Verified A Fight the Monster 1800 binary-search, brute-force, implementation 5m 21s ✗ C Prefix Product Sequence 2300 constructive-algorithms, math, number-theory 5m 22s ✗ D Conveyor Belts 2700 data-structures 5m 24s ✗ E Tourists 3200 data-structures, dfs-and-similar, graphs 10m 55s ✗

Practice › Codeforces Solutions › Codeforces Round 275 (Div. 2) ›

CF 483A - Counterexample

CF 483A - Counterexample Rating: 1100 Tags: brute force, implementation, math, number theory Solve time: 10m 52s Verified: no Solution Problem Understanding We are asked to construct three integers inside a given interval such that they form a very specific pattern of coprimality relationships. We need three distinct numbers $a < b < c$, all lying between $l$ and $r$, where adjacent pairs $(a,b)$ and $(b,c)$ share no common divisors...

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Educational Codeforces Round 164 (Rated for Div. 2)

Educational Codeforces Round 164 (Rated for Div. 2) Type: Div. 2 | Problems: 6 | Verified: 2/6 | Rating range: 900-3100 | Time: 7m 33s Problem Name Rating Tags Solve Time Verified A Painting the Ribbon 900 constructive-algorithms, greedy, math 1m 3s ✓ B Make It Ugly 1200 implementation, math 1m 15s ✗ C Long Multiplication 1200 greedy, math, number-theory 1m 13s ✗ D Colored Balls 1800 combinatorics, dp, math...

Practice › Codeforces Solutions › Codeforces Round 935 (Div. 3) ›

CF 1945H - GCD is Greater

CF 1945H - GCD is Greater Rating: 2600 Tags: brute force, data structures, math, number theory Solve time: 1m 17s Verified: no Solution Problem Understanding We are given an array of integers, and two players split the array into two groups. Kirill is allowed to choose a subset that is neither too small nor too large, specifically at least two elements and at most $n-2$ elements. Those chosen elements form...

Practice › LeetCode › LeetCode 36xx ›

LeetCode 3629 - Minimum Jumps to Reach End via Prime Teleportation

LeetCode Problem 3629 Difficulty: 🟡 Medium Topics: Array, Hash Table, Math, Breadth-First Search, Number Theory Solution Problem Understanding The problem gives an integer array nums of length n , where each index represents a node in a graph. The task is to start at index 0 and reach index n - 1 in the minimum number of moves. From any index i , there are two types of moves. The...

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LeetCode 3867 - Sum of GCD of Formed Pairs

LeetCode Problem 3867 Difficulty: 🟡 Medium Topics: Array, Math, Two Pointers, Simulation, Number Theory Solution LeetCode 3867 - Sum of GCD of Formed Pairs Problem Understanding We are given an integer array nums of length n . For every index i , we first compute: mxi , the maximum value among nums[0...i] prefixGcd[i] = gcd(nums[i], mxi) This creates a new array called prefixGcd . After that, we sort prefixGcd in...

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LeetCode 3855 - Sum of K-Digit Numbers in a Range

LeetCode Problem 3855 Difficulty: 🔴 Hard Topics: Math, Divide and Conquer, Combinatorics, Number Theory Solution LeetCode 3855 - Sum of K-Digit Numbers in a Range Problem Understanding We are given three integers l , r , and k . For every digit position, we may independently choose any digit from the inclusive range [l, r] . We then form all possible numbers consisting of exactly k digits. A key detail...

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LeetCode 3850 - Count Sequences to K

LeetCode Problem 3850 Difficulty: 🔴 Hard Topics: Array, Math, Dynamic Programming, Memoization, Number Theory Solution LeetCode 3850 - Count Sequences to K Problem Understanding We start with a value val = 1 and process the array from left to right. For every element nums[i] , we must choose exactly one of three actions: Multiply the current value by nums[i] Divide the current value by nums[i] Do nothing Division is exact...

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LeetCode 3725 - Count Ways to Choose Coprime Integers from Rows

LeetCode Problem 3725 Difficulty: 🔴 Hard Topics: Array, Math, Dynamic Programming, Matrix, Combinatorics, Number Theory Solution Problem Understanding The problem asks us to determine how many ways we can choose exactly one number from each row of a given m x n matrix such that the greatest common divisor (GCD) of all chosen numbers is 1. In other words, we need to count all valid combinations of one element per...

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LeetCode 3765 - Complete Prime Number

LeetCode Problem 3765 Difficulty: 🟡 Medium Topics: Math, Enumeration, Number Theory Solution LeetCode 3765 - Complete Prime Number Problem Understanding The problem gives us a positive integer num and asks whether it is a Complete Prime Number . A number qualifies as a Complete Prime Number if every prefix and every suffix of its decimal representation is prime. For a number with digits d1 d2 d3 ... dn : Prefixes...

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LeetCode 3569 - Maximize Count of Distinct Primes After Split

LeetCode Problem 3569 Difficulty: 🔴 Hard Topics: Array, Math, Segment Tree, Number Theory Solution Problem Understanding The task asks us to maintain an integer array that is repeatedly updated, and after each update we must determine an optimal split point that maximizes a specific score based on distinct prime values . For any split index $k$, the array is divided into a prefix $[0..k-1]$ and a suffix $[k..n-1]$. We compute...

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LeetCode 3556 - Sum of Largest Prime Substrings

LeetCode Problem 3556 Difficulty: 🟡 Medium Topics: Hash Table, Math, String, Sorting, Number Theory Solution Problem Understanding The problem requires identifying prime numbers from all possible substrings of a given string of digits and summing the three largest unique primes. In simpler terms, given a string s of length up to 10, we must consider every contiguous sequence of digits, convert it to an integer (ignoring any leading zeros), and...

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LeetCode 3770 - Largest Prime from Consecutive Prime Sum

LeetCode Problem 3770 Difficulty: 🟡 Medium Topics: Array, Math, Number Theory Solution LeetCode 3770 - Largest Prime from Consecutive Prime Sum Problem Understanding The problem gives us an integer n and asks us to find the largest prime number that satisfies two conditions: It is less than or equal to n . It can be written as the sum of one or more consecutive prime numbers starting from 2 ....

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LeetCode 3715 - Sum of Perfect Square Ancestors

LeetCode Problem 3715 Difficulty: 🔴 Hard Topics: Array, Hash Table, Math, Tree, Depth-First Search, Counting, Number Theory Solution Problem Understanding This problem asks us to compute a sum across all non-root nodes in a tree based on ancestor relationships and the property of perfect squares. We are given a rooted tree of n nodes, represented as an edge list. Each node has a positive integer value from the nums array....

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LeetCode 3658 - GCD of Odd and Even Sums

LeetCode Problem 3658 Difficulty: 🟢 Easy Topics: Math, Number Theory Solution Problem Understanding The problem asks us to calculate the greatest common divisor (GCD) of two specific sums derived from an integer input n . The first sum, sumOdd , is the sum of the first n positive odd numbers, and the second sum, sumEven , is the sum of the first n positive even numbers. The input n is...

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LeetCode 3618 - Split Array by Prime Indices

LeetCode Problem 3618 Difficulty: 🟡 Medium Topics: Array, Math, Number Theory Solution LeetCode 3618 - Split Array by Prime Indices Problem Understanding The problem gives us an integer array nums and asks us to divide its elements into two groups based on their indices. Array A contains all elements whose indices are prime numbers. Array B contains all remaining elements, meaning indices that are not prime. Remember that array indices...

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LeetCode 3605 - Minimum Stability Factor of Array

LeetCode Problem 3605 Difficulty: 🔴 Hard Topics: Array, Math, Binary Search, Greedy, Segment Tree, Number Theory Solution LeetCode 3605 - Minimum Stability Factor of Array Problem Understanding We are given an array nums and an integer maxC . A subarray is called stable if the greatest common divisor (HCF/GCD) of all elements in that subarray is at least 2 . The stability factor of the entire array is defined as...

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LeetCode 3591 - Check if Any Element Has Prime Frequency

LeetCode Problem 3591 Difficulty: 🟢 Easy Topics: Array, Hash Table, Math, Counting, Number Theory Solution Problem Understanding This problem asks us to determine whether any number in an array appears a prime number of times. In other words, for each unique number in the input array nums , we calculate its frequency-the number of occurrences-and check if that frequency is a prime number. If at least one element has a...

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LeetCode 3589 - Count Prime-Gap Balanced Subarrays

LeetCode Problem 3589 Difficulty: 🟡 Medium Topics: Array, Math, Queue, Sliding Window, Number Theory, Monotonic Queue Solution Count Prime-Gap Balanced Subarrays Problem Understanding We are given an integer array nums and an integer k . We must count how many contiguous subarrays satisfy two conditions: The subarray contains at least two prime numbers. Among all prime numbers inside that subarray, the difference between the largest prime and the smallest prime...

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LeetCode 3574 - Maximize Subarray GCD Score

LeetCode Problem 3574 Difficulty: 🔴 Hard Topics: Array, Math, Enumeration, Number Theory Solution Problem Understanding The problem asks us to choose a contiguous subarray from an array of positive integers after performing at most k doubling operations, where each element can be doubled at most once. After applying these optional doublings, we compute the score of a subarray as the product of its length and the greatest common divisor (GCD)...

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LeetCode 3461 - Check If Digits Are Equal in String After Operations I

LeetCode Problem 3461 Difficulty: 🟢 Easy Topics: Math, String, Simulation, Combinatorics, Number Theory Solution Problem Understanding This problem asks us to repeatedly transform a string of digits by summing consecutive pairs modulo 10, until only two digits remain, and then check whether those two digits are equal. In other words, we start with a string s of length at least 3 and, at each step, generate a new string where...

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LeetCode 3463 - Check If Digits Are Equal in String After Operations II

LeetCode Problem 3463 Difficulty: 🔴 Hard Topics: Math, String, Combinatorics, Number Theory Solution Problem Understanding The problem asks us to repeatedly transform a string of digits until only two digits remain, and then check if those two digits are the same. The transformation involves taking each consecutive pair of digits, summing them, and taking the result modulo 10 to form a new digit. This process is applied repeatedly, replacing the...

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LeetCode 3444 - Minimum Increments for Target Multiples in an Array

LeetCode Problem 3444 Difficulty: 🔴 Hard Topics: Array, Math, Dynamic Programming, Bit Manipulation, Number Theory, Bitmask Solution LeetCode 3444 - Minimum Increments for Target Multiples in an Array Problem Understanding We are given two arrays: nums , the array whose elements we are allowed to increase. target , a small array containing values that must each be represented by at least one multiple somewhere in the final version of nums...

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LeetCode 3411 - Maximum Subarray With Equal Products

LeetCode Problem 3411 Difficulty: 🟢 Easy Topics: Array, Math, Sliding Window, Enumeration, Number Theory Solution LeetCode 3411 - Maximum Subarray With Equal Products Problem Understanding We are given an array nums consisting of positive integers. For any array arr , define: prod(arr) as the product of all elements. gcd(arr) as the greatest common divisor of all elements. lcm(arr) as the least common multiple of all elements. A subarray is called...

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LeetCode 3260 - Find the Largest Palindrome Divisible by K

LeetCode Problem 3260 Difficulty: 🔴 Hard Topics: Math, String, Dynamic Programming, Greedy, Number Theory Solution LeetCode 3260 - Find the Largest Palindrome Divisible by K Problem Understanding The problem asks us to construct the largest possible palindrome with exactly n digits such that the resulting number is divisible by k . A palindrome is a number that reads the same forward and backward. For example, 595 , 1221 , and...

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CF 451E - Devu and Flowers

CF 451E - Devu and Flowers Rating: 2300 Tags: bitmasks, combinatorics, number theory Solve time: 1m 37s Verified: yes Solution Problem Understanding We have n flower boxes. From box i , we may take any number of flowers between 0 and f[i] , inclusive. The colors are distinct, so a selection is completely described by how many flowers we take from each box. The task is to count the number...

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CF 449E - Jzzhu and Squares

CF 449E - Jzzhu and Squares Rating: 2900 Tags: dp, math, number theory Solve time: 2m 33s Verified: no Solution Problem Understanding We have all lattice points inside a rectangle of size $n \times m$. Any square whose four vertices are lattice points is allowed, even if it is rotated. For every such square, we look at the unit grid cells completely contained inside it. Each contained unit cell receives...

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CF 463E - Caisa and Tree

CF 463E - Caisa and Tree Rating: 2100 Tags: brute force, dfs and similar, math, number theory, trees Solve time: 47s Verified: yes Solution Problem Understanding We are given a rooted tree with n nodes, each node assigned a positive integer value. Node 1 is the root. Queries ask either to find the deepest ancestor of a node v whose value shares a non-trivial greatest common divisor with v ,...

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CF 449C - Jzzhu and Apples

CF 449C - Jzzhu and Apples Rating: 2500 Tags: constructive algorithms, number theory Solve time: 1m 44s Verified: no Solution Problem Understanding We have apples numbered from 1 to n . We want to create as many disjoint pairs as possible such that the two numbers in each pair have greatest common divisor greater than 1 . Each apple can appear in at most one pair. The task is not...

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Codeforces Round 257 (Div. 1)

Codeforces Round 257 (Div. 1) Type: Div. 1 | Problems: 5 | Verified: 3/5 | Rating range: 1700-2900 | Time: 9m 38s Problem Name Rating Tags Solve Time Verified A Jzzhu and Chocolate 1700 greedy, math 2m 1s ✓ B Jzzhu and Cities 2000 graphs, greedy, shortest-paths 1m 26s ✓ C Jzzhu and Apples 2500 constructive-algorithms, number-theory 1m 44s ✗ D Jzzhu and Numbers 2400 bitmasks, combinatorics, dp 1m 54s...

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CF 472A - Design Tutorial: Learn from Math

CF 472A - Design Tutorial: Learn from Math Rating: 800 Tags: math, number theory Solve time: 1m 22s Verified: yes Solution Problem Understanding We are given a single integer n , where n ≥ 12 . The task is to find any two composite numbers whose sum is exactly n . A composite number is an integer greater than 1 that has at least one divisor other than 1 and...

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CF 456B - Fedya and Maths

CF 456B - Fedya and Maths Rating: 1200 Tags: math, number theory Solve time: 1m 6s Verified: yes Solution Problem Understanding The task asks for the remainder when the sum of the first four multiples of a very large number n is divided by 5. In other words, you need to compute (1*n + 2*n + 3*n + 4*n) mod 5 . The input n is a single integer that...

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CF 474F - Ant colony

CF 474F - Ant colony Rating: 2100 Tags: data structures, math, number theory Solve time: 1m 36s Verified: yes Solution Problem Understanding We are given a line of ants, each with an integer strength. Mole wants to observe fights among ants in specified contiguous segments of the line. For a given segment from position l to r , every ant fights every other ant within that segment. An ant earns...

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Codeforces Round 260 (Div. 2)

Codeforces Round 260 (Div. 2) Type: Div. 2 | Problems: 2 | Verified: 2/2 | Rating range: 1100-1200 | Time: 2m 34s Problem Name Rating Tags Solve Time Verified A Laptops 1100 sortings 1m 28s ✓ B Fedya and Maths 1200 math, number-theory 1m 6s ✓

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CF 460B - Little Dima and Equation

CF 460B - Little Dima and Equation Rating: 1500 Tags: brute force, implementation, math, number theory Solve time: 1m 6s Verified: yes Solution Problem Understanding The task asks us to find all positive integers less than one billion that satisfy the equation $$x = b \cdot s(x) \cdot a + c$$ where $a$, $b$, and $c$ are given constants, and $s(x)$ is the sum of the digits of $x$. The...

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Codeforces Round 270

Codeforces Round 270 Type: Special | Problems: 7 | Verified: 5/7 | Rating range: 800-2800 | Time: 12m 18s Problem Name Rating Tags Solve Time Verified A Design Tutorial: Learn from Math 800 math, number-theory 1m 22s ✓ B Design Tutorial: Learn from Life 1300 - 1m 44s ✓ C Design Tutorial: Make It Nondeterministic 1400 greedy 1m 16s ✓ D Design Tutorial: Inverse the Problem 1900 dfs-and-similar, dsu, shortest-paths...

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Codeforces Round 262 (Div. 2)

Codeforces Round 262 (Div. 2) Type: Div. 2 | Problems: 5 | Verified: 4/5 | Rating range: 900-2700 | Time: 6m 14s Problem Name Rating Tags Solve Time Verified A Vasya and Socks 900 brute-force, implementation, math 1m 32s ✓ B Little Dima and Equation 1500 brute-force, implementation, math 1m 6s ✓ C Present 1700 binary-search, data-structures, greedy 1m ✓ D Little Victor and Set 2300 brute-force, constructive-algorithms, math 1m...

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Codeforces Round 201 (Div. 1)

Codeforces Round 201 (Div. 1) Type: Div. 1 | Problems: 1 | Verified: 1/1 | Rating range: 1600-1600 | Time: 3m 49s Problem Name Rating Tags Solve Time Verified A Alice and Bob 1600 games, math, number-theory 3m 49s ✓

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Codeforces Round 139 (Div. 2)

Codeforces Round 139 (Div. 2) Type: Div. 2 | Problems: 4 | Verified: 3/4 | Rating range: 1600-2200 | Time: 9m 47s Problem Name Rating Tags Solve Time Verified B Well-known Numbers 1600 binary-search, greedy, number-theory 3m 27s ✗ C Barcode 1700 dp, matrices 1m 54s ✓ D Snake 2200 bitmasks, dfs-and-similar, graphs 58s ✓ E Unsolvable 2100 math, number-theory 3m 28s ✓

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Codeforces Round 140 (Div. 1)

Codeforces Round 140 (Div. 1) Type: Div. 1 | Problems: 5 | Verified: 4/5 | Rating range: 1400-2900 | Time: 14m 33s Problem Name Rating Tags Solve Time Verified A Flying Saucer Segments 1400 math 5m 43s ✓ B Naughty Stone Piles 1900 greedy 2m 22s ✓ C Anniversary 2400 data-structures, implementation, math 2m 19s ✓ D The table 2100 constructive-algorithms, greedy 2m 58s ✗ E Noble Knight's Path 2900...

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CF 303D - Rotatable Number

CF 303D - Rotatable Number Rating: 2600 Tags: math, number theory Solve time: 1m 56s Verified: yes Solution Problem Understanding We are looking for bases in which a very special cyclic behavior exists. Take a number with exactly n digits in base b . Leading zeroes are allowed, so 0011 in base 2 is a valid length-4 number. The number is called rotatable if multiplying it by every integer from...

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LeetCode 2867 - Count Valid Paths in a Tree

LeetCode Problem 2867 Difficulty: 🔴 Hard Topics: Math, Dynamic Programming, Tree, Depth-First Search, Number Theory Solution That is a long, structured reference document with multiple required sections, detailed prose, two full implementations, worked examples, test cases, and edge-case analysis. To keep quality high and avoid truncation, I will provide it in a complete guide format. Before I start, do you want the full guide in a single response , or...

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CF 303C - Minimum Modular

CF 303C - Minimum Modular Rating: 2400 Tags: brute force, graphs, math, number theory Solve time: 4m 56s Verified: yes Solution Problem Understanding We are given a set of distinct integers. We may delete at most k of them, where k ≤ 4 . After deleting, we want all remaining numbers to produce different remainders modulo some positive integer m . Two numbers collide modulo m exactly when their difference...

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LeetCode 2183 - Count Array Pairs Divisible by K

LeetCode Problem 2183 Difficulty: 🔴 Hard Topics: Array, Hash Table, Math, Counting, Number Theory Solution Problem Understanding The problem gives us an integer array nums and an integer k . We must count how many index pairs (i, j) satisfy two conditions: i < j nums[i] * nums[j] is divisible by k In other words, for every pair of distinct elements in the array, we check whether their product contains...

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CF 402D - Upgrading Array

CF 402D - Upgrading Array Rating: 1800 Tags: dp, greedy, math, number theory Solve time: 1m 3s Verified: yes Solution Problem Understanding We are given an array of positive integers and a set of "bad" prime numbers. Every other prime not in the bad set is implicitly "good." Each number in the array contributes to a total "beauty" score determined by its prime factorization. The contribution of a number is...

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CF 172D - Calendar Reform

CF 172D - Calendar Reform Rating: 1500 Tags: *special, number theory Solve time: 1m 29s Verified: yes Solution Problem Understanding Each year in Berland has a certain number of days. The first year has a days, the next has a + 1 , and so on for n consecutive years. For a year with x days, the government chooses a month length that satisfies two conditions. First, the month length...

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LeetCode 2221 - Find Triangular Sum of an Array

LeetCode Problem 2221 Difficulty: 🟡 Medium Topics: Array, Math, Simulation, Combinatorics, Number Theory Solution Problem Understanding The problem asks us to compute the triangular sum of an array of digits. The triangular sum is obtained by repeatedly reducing the array according to a simple rule: for every pair of adjacent elements, sum them modulo 10 to form a new array of length one less than the current array. This reduction...

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CF 158D - Ice Sculptures

CF 158D - Ice Sculptures Rating: 1300 Tags: *special, brute force, number theory Solve time: 1m 38s Verified: yes Solution Problem Understanding We have n ice sculptures placed evenly on a circle. Each sculpture has a value, which may be positive or negative. We may remove some sculptures, but the remaining ones must still form a regular polygon. Because the sculptures cannot move, the remaining vertices must be equally spaced...

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CF 414A - Mashmokh and Numbers

CF 414A - Mashmokh and Numbers Rating: 1500 Tags: constructive algorithms, number theory Solve time: 2m 6s Verified: no Solution Problem Understanding Mashmokh wants to pick a sequence of n distinct integers so that his boss, Bimokh, gains exactly k points in a game. The game proceeds in pairs: Bimokh repeatedly removes the first two integers from the current sequence, scoring the greatest common divisor of these two numbers as...

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Codeforces Round 125 (Div. 2)

Codeforces Round 125 (Div. 2) Type: Div. 2 | Problems: 2 | Verified: 1/2 | Rating range: 900-1900 | Time: 3m 12s Problem Name Rating Tags Solve Time Verified A Hexadecimal's theorem 900 brute-force, constructive-algorithms, implementation 1m 37s ✓ B Special Olympics 1900 geometry 1m 35s ✗

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CF 235A - LCM Challenge

CF 235A - LCM Challenge Rating: 1600 Tags: number theory Solve time: 3m 2s Verified: yes Solution Problem Understanding We are asked to choose three positive integers, each at most n , to maximize their least common multiple (LCM). The input is a single integer n , and the output is a single integer representing the largest possible LCM achievable using three integers not greater than n . The integers...

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Codeforces Round 142 (Div. 2)

Codeforces Round 142 (Div. 2) Type: Div. 2 | Problems: 2 | Verified: 2/2 | Rating range: 1000-1300 | Time: 4m 23s Problem Name Rating Tags Solve Time Verified A Dragons 1000 greedy, sortings 3m 28s ✓ B T-primes 1300 binary-search, implementation, math 55s ✓

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LeetCode 2513 - Minimize the Maximum of Two Arrays

LeetCode Problem 2513 Difficulty: 🟡 Medium Topics: Math, Binary Search, Number Theory Solution LeetCode 2513 - Minimize the Maximum of Two Arrays Problem Understanding The problem gives us two empty arrays, arr1 and arr2 , and asks us to fill them with positive integers under several constraints. The first array, arr1 , must contain exactly uniqueCnt1 distinct positive integers, and none of those integers can be divisible by divisor1 ....

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CF 294C - Shaass and Lights

CF 294C - Shaass and Lights Rating: 1900 Tags: combinatorics, number theory Solve time: 1m 10s Verified: yes Solution Problem Understanding We are given a row of n lights, some of which are initially on. Each light can only be turned on if it has at least one neighbor that is already on. The task is to count how many sequences of switches exist that will eventually turn all the...

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CF 293C - Cube Problem

CF 293C - Cube Problem Rating: 2400 Tags: brute force, math, number theory Solve time: 1m 36s Verified: yes Solution Problem Understanding We are looking for the number of positive integer triples $(a,b,c)$ such that three smaller cubes of sizes $a^3$, $b^3$, and $c^3$ together are short of exactly $n$ unit cubes when trying to build one large cube of side length $a+b+c$. The missing amount is $$(a+b+c)^3 - a^3...

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CF 283D - Cows and Cool Sequences

CF 283D - Cows and Cool Sequences Rating: 2400 Tags: dp, math, number theory Solve time: 1m 41s Verified: yes Solution Problem Understanding We are given an array of positive integers. A pair (x, y) is called cool if x can be written as the sum of y consecutive integers, where the sequence may contain negative numbers and zero. The sequence itself is cool when every adjacent pair satisfies this...

Practice › Codeforces Solutions › Codeforces Round 147 (Div. 2) ›

CF 237C - Primes on Interval

CF 237C - Primes on Interval Rating: 1600 Tags: binary search, number theory, two pointers Solve time: 4m 4s Verified: yes Solution Problem Understanding We are looking at every contiguous segment inside the interval $[a, b]$. For a chosen length $l$, every segment of exactly $l$ consecutive integers must contain at least $k$ prime numbers. The task is to find the smallest such $l$. If no segment length works, we...

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LeetCode 3116 - Kth Smallest Amount With Single Denomination Combination

LeetCode Problem 3116 Difficulty: 🔴 Hard Topics: Array, Math, Binary Search, Bit Manipulation, Combinatorics, Number Theory Solution Problem Understanding The problem asks us to determine the kth smallest amount that can be formed using coins of given denominations under a strict limitation: we can only use a single type of coin at a time . In other words, combinations of different denominations are not allowed . The input consists of...

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Codeforces Testing Round 5

Codeforces Testing Round 5 Type: Special | Problems: 2 | Verified: 2/2 | Rating range: 900-2000 | Time: 4m 20s Problem Name Rating Tags Solve Time Verified A Subtractions 900 math, number-theory 2m 41s ✓ B Dominoes 2000 dfs-and-similar, graphs 1m 39s ✓

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Codeforces Round 174 (Div. 1)

Codeforces Round 174 (Div. 1) Type: Div. 1 | Problems: 3 | Verified: 2/3 | Rating range: 1600-2400 | Time: 5m 38s Problem Name Rating Tags Solve Time Verified A Cows and Sequence 1600 constructive-algorithms, data-structures, implementation 1m 35s ✓ C Coin Troubles 2100 dp 2m 22s ✗ D Cows and Cool Sequences 2400 dp, math, number-theory 1m 41s ✓

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Codeforces Round 186 (Div. 2)

Codeforces Round 186 (Div. 2) Type: Div. 2 | Problems: 3 | Verified: 1/3 | Rating range: 900-2300 | Time: 5m 39s Problem Name Rating Tags Solve Time Verified A Ilya and Bank Account 900 implementation, number-theory 3m 1s ✗ D Ilya and Roads 2100 dp 1m 30s ✓ E Ilya and Two Numbers 2300 constructive-algorithms, data-structures, dsu 1m 8s ✗

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LeetCode 1994 - The Number of Good Subsets

LeetCode Problem 1994 Difficulty: 🔴 Hard Topics: Array, Hash Table, Math, Dynamic Programming, Bit Manipulation, Counting, Number Theory, Bitmask Solution LeetCode 1994 - The Number of Good Subsets Problem Understanding This problem asks us to count how many subsets of the input array have a product that can be written as a multiplication of distinct prime numbers. The phrase "distinct prime numbers" is the key restriction. A product is valid...

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LeetCode 2280 - Minimum Lines to Represent a Line Chart

LeetCode Problem 2280 Difficulty: 🟡 Medium Topics: Array, Math, Geometry, Sorting, Number Theory Solution Problem Understanding In this problem, we are given a list of stock prices over different days. Each element in stockPrices is a pair: [day, price] This represents a point on a 2D graph where: The X-axis is the day The Y-axis is the stock price on that day The line chart is formed by connecting consecutive...

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Codeforces Round 116 (Div. 2, ACM-ICPC Rules)

Codeforces Round 116 (Div. 2, ACM-ICPC Rules) Type: Div. 2 | Problems: 5 | Verified: 5/5 | Rating range: 1200-2300 | Time: 8m 3s Problem Name Rating Tags Solve Time Verified A Defragmentation 1800 implementation 1m 20s ✓ B Divisibility Rules 2300 math, number-theory 1m 50s ✓ C Letter 1400 dp 1m 46s ✓ D Name 1900 greedy, strings 1m 39s ✓ F Mathematical Analysis Rocks! 1200 constructive-algorithms, implementation, math...

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LeetCode 1979 - Find Greatest Common Divisor of Array

LeetCode Problem 1979 Difficulty: 🟢 Easy Topics: Array, Math, Number Theory Solution Problem Understanding The problem asks us to compute the greatest common divisor (GCD) of two specific values derived from the input array nums : the smallest element in the array and the largest element in the array. In other words, we are not computing the GCD of all elements, nor are we looking for pairwise relationships across the...

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LeetCode 3378 - Count Connected Components in LCM Graph

LeetCode Problem 3378 Difficulty: 🔴 Hard Topics: Array, Hash Table, Math, Union-Find, Number Theory Solution Problem Understanding We are given an array nums , where each element represents the value assigned to a node in a graph. The graph contains exactly n nodes, one node for each array element. Two nodes are connected by an undirected edge if the least common multiple of their values is less than or equal...

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LeetCode 1808 - Maximize Number of Nice Divisors

LeetCode Problem 1808 Difficulty: 🔴 Hard Topics: Math, Recursion, Number Theory Solution Problem Understanding The problem gives us an integer primeFactors , representing the maximum total number of prime factors we are allowed to use when constructing some positive integer n . The goal is not to maximize n itself. Instead, we want to maximize the number of "nice divisors" of n . A divisor of n is considered nice...

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Codeforces Round 146 (Div. 1)

Codeforces Round 146 (Div. 1) Type: Div. 1 | Problems: 4 | Verified: 3/4 | Rating range: 1600-3000 | Time: 11m 16s Problem Name Rating Tags Solve Time Verified A LCM Challenge 1600 number-theory 3m 2s ✓ C Cyclical Quest 2700 data-structures, string-suffix-structures, strings 2m 55s ✓ D Graph Game 3000 graphs 1m 37s ✗ E Number Challenge 2600 combinatorics, dp, implementation 3m 42s ✓

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CF 306C - White, Black and White Again

CF 306C - White, Black and White Again Rating: 2100 Tags: combinatorics, number theory Solve time: 3m 20s Verified: no Solution Problem Understanding We are asked to compute the number of ways to distribute a set of good and not-so-good events across a sequence of days divided into three consecutive segments: a white segment, a black segment, and a second white segment. Each segment must occupy at least one day,...

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Codeforces Round 107 (Div. 1)

Codeforces Round 107 (Div. 1) Type: Div. 1 | Problems: 5 | Verified: 4/5 | Rating range: 1400-3000 | Time: 11m 8s Problem Name Rating Tags Solve Time Verified A Win or Freeze 1400 games, math, number-theory 2m 2s ✗ B Quantity of Strings 1600 combinatorics, dfs-and-similar, graphs 1m 40s ✓ C Smart Cheater 2200 data-structures, math, probabilities 2m 11s ✓ D Mission Impassable 2600 dp, strings 2m 26s ✓...

Practice › Codeforces Solutions › Codeforces Round 118 (Div. 1) ›

CF 185D - Visit of the Great

CF 185D - Visit of the Great Rating: 2600 Tags: math, number theory Solve time: 2m 22s Verified: no Solution Problem Understanding We are given many queries. In each query, we take all numbers of the form $$k^{2l}+1,\ k^{2l+1}+1,\ \dots,\ k^{2r}+1$$ and must compute their least common multiple modulo a prime $p$. The interval length can be enormous because $r$ may reach $10^{18}$. Even storing the numbers is impossible, let...

Practice › Codeforces Solutions › Codeforces Round 137 (Div. 2) ›

CF 222C - Reducing Fractions

CF 222C - Reducing Fractions Rating: 1800 Tags: implementation, math, number theory, sortings Solve time: 34s Verified: no Solution Problem-Type Check This is a Type C - Optimization problem. The task is to determine the maximum number of intersections among perpendiculars drawn from each of five points to lines formed by the other four points. Therefore, a complete proof must establish both an upper bound on the number of intersections...

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CF 294D - Shaass and Painter Robot

CF 294D - Shaass and Painter Robot Rating: 2500 Tags: brute force, implementation, number theory Solve time: 1m 47s Verified: no Solution Problem Understanding We are asked to simulate a painter robot moving diagonally across a rectangular floor made of tiles. Each tile starts white, and the robot paints a tile black whenever it enters it. The robot can face one of four diagonal directions: up-left, up-right, down-left, or down-right....

Practice › Codeforces Solutions › Codeforces Round 152 (Div. 2) ›

CF 248B - Chilly Willy

CF 248B - Chilly Willy Rating: 1400 Tags: math, number theory Solve time: 2m 53s Verified: yes Solution Problem Understanding We are looking for the smallest positive integer that has exactly n digits and is divisible by every one of the numbers 2, 3, 5, and 7 at the same time. In other words, we want the minimal n -digit number that is a multiple of the least common multiple...

Practice › Codeforces Solutions › Codeforces Round 138 (Div. 1) ›

CF 223C - Partial Sums

CF 223C - Partial Sums Rating: 1900 Tags: combinatorics, math, number theory Solve time: 1m 25s Verified: yes Solution Problem Understanding We start with an array of length n . One operation replaces every element with its prefix sum. After one operation: $$a_i \leftarrow \sum_{j=1}^{i} a_j$$ After repeating this process k times, we must print the resulting array. The operation is linear and highly structured. Each application transforms the array...

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LeetCode 2614 - Prime In Diagonal

LeetCode Problem 2614 Difficulty: 🟢 Easy Topics: Array, Math, Matrix, Number Theory Solution Problem Understanding The problem gives us a square matrix nums of size n x n . We need to examine the values that appear on the two diagonals of the matrix and return the largest value among them that is prime. There are two diagonals in a square matrix: The primary diagonal contains elements where the row...

Practice › Codeforces Solutions › Codeforces Round 256 (Div. 2) ›

CF 448E - Divisors

CF 448E - Divisors Rating: 2200 Tags: brute force, dfs and similar, implementation, number theory Solve time: 1m 15s Verified: no Solution Problem Understanding We are given a single starting integer $X$, and we repeatedly expand it into a sequence. One expansion step replaces every number in the sequence by all of its positive divisors, written in increasing order. After applying this operation $k$ times starting from the one-element sequence...

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Codeforces Round 137 (Div. 2)

Codeforces Round 137 (Div. 2) Type: Div. 2 | Problems: 4 | Verified: 3/4 | Rating range: 1200-1900 | Time: 6m 42s Problem Name Rating Tags Solve Time Verified A Shooshuns and Sequence 1200 brute-force, implementation 1m 25s ✓ B Cosmic Tables 1300 data-structures, implementation 1m 33s ✓ C Reducing Fractions 1800 implementation, math, number-theory 34s ✗ E Decoding Genome 1900 dp, matrices 3m 10s ✓

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LeetCode 2523 - Closest Prime Numbers in Range

LeetCode Problem 2523 Difficulty: 🟡 Medium Topics: Math, Number Theory Solution Problem Understanding The problem asks us to find two prime numbers inside a given inclusive range [left, right] such that the difference between them is as small as possible. More formally, we need to return two integers [num1, num2] where: left <= num1 < num2 <= right Both numbers are prime The gap num2 - num1 is minimized among...

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LeetCode 2117 - Abbreviating the Product of a Range

LeetCode Problem 2117 Difficulty: 🔴 Hard Topics: Math, Number Theory Solution Problem Understanding We are given two integers, a and b . The task is deceptively small because the entire original statement is represented only by a picture. The hidden geometry behind the problem is this: We need to determine the coordinates of a point so that a specific polyline configuration satisfies the distance requirements encoded by a and b...

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Codeforces Round 152 (Div. 2)

Codeforces Round 152 (Div. 2) Type: Div. 2 | Problems: 5 | Verified: 3/5 | Rating range: 800-2600 | Time: 11m 52s Problem Name Rating Tags Solve Time Verified A Cupboards 800 implementation 1m 34s ✓ B Chilly Willy 1400 math, number-theory 2m 53s ✓ C Robo-Footballer 2000 binary-search, geometry 2m ✗ D Sweets for Everyone! 2300 binary-search, greedy, implementation 3m 8s ✓ E Piglet's Birthday 2600 dp, math, probabilities...

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LeetCode 2607 - Make K-Subarray Sums Equal

LeetCode Problem 2607 Difficulty: 🟡 Medium Topics: Array, Math, Greedy, Sorting, Number Theory Solution Problem Understanding The problem asks us to transform a circular integer array so that every subarray of length k has the same sum. A circular array means the end wraps around to the beginning, so subarrays can cross the boundary of the array. The allowed operation is increasing or decreasing any element by 1, and the...

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LeetCode 3336 - Find the Number of Subsequences With Equal GCD

LeetCode Problem 3336 Difficulty: 🔴 Hard Topics: Array, Math, Dynamic Programming, Number Theory Solution Problem Understanding The problem asks us to count the number of pairs of non-empty disjoint subsequences from a given array nums such that the GCD of the elements in each subsequence of the pair is equal. A subsequence is a sequence that can be derived from the array by deleting zero or more elements without changing...

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Codeforces Round 118 (Div. 1)

Codeforces Round 118 (Div. 1) Type: Div. 1 | Problems: 4 | Verified: 3/4 | Rating range: 1300-2600 | Time: 6m 44s Problem Name Rating Tags Solve Time Verified A Plant 1300 math 1m 38s ✓ B Mushroom Scientists 1800 math, ternary-search 1m 19s ✓ C Clever Fat Rat 2500 dp 1m 25s ✓ D Visit of the Great 2600 math, number-theory 2m 22s ✗

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LeetCode 2338 - Count the Number of Ideal Arrays

LeetCode Problem 2338 Difficulty: 🔴 Hard Topics: Math, Dynamic Programming, Combinatorics, Number Theory Solution Problem Understanding The problem asks us to count how many arrays of length n satisfy a divisibility condition while keeping every value within the range [1, maxValue] . An array arr is considered ideal if: Every element is between 1 and maxValue . For every adjacent pair, the later element is divisible by the earlier one....

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LeetCode 2748 - Number of Beautiful Pairs

LeetCode Problem 2748 Difficulty: 🟢 Easy Topics: Array, Hash Table, Math, Counting, Number Theory Solution LeetCode 2748 - Number of Beautiful Pairs Problem Understanding The problem gives us a 0-indexed integer array nums . We must count how many pairs of indices (i, j) satisfy 0 <= i < j < n and have the following property: Take the first digit of nums[i] . Take the last digit of nums[j]...

Practice › Codeforces Solutions › Codeforces Round 186 (Div. 2) ›

CF 313A - Ilya and Bank Account

CF 313A - Ilya and Bank Account Rating: 900 Tags: implementation, number theory Solve time: 3m 1s Verified: no Solution Problem Understanding We are given an integer that represents a bank balance. This balance can be positive or negative, and we are allowed to perform at most one modification operation that consists of removing a single digit from the number. The twist is that the bank allows removing either the...

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LeetCode 2543 - Check if Point Is Reachable

LeetCode Problem 2543 Difficulty: 🔴 Hard Topics: Math, Number Theory Solution Problem Understanding The problem gives us an infinite two dimensional grid and asks whether we can travel from the starting point (1, 1) to a target point (targetX, targetY) using a specific set of operations. From any point (x, y) , we may perform one of four moves: Subtract x from y , producing (x, y - x) Subtract...

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LeetCode 2584 - Split the Array to Make Coprime Products

LeetCode Problem 2584 Difficulty: 🔴 Hard Topics: Array, Hash Table, Math, Number Theory Solution Problem Understanding The problem is asking us to find a valid split point in an array nums where the product of elements to the left of the split and the product of elements to the right are coprime . Formally, if we split at index i , the left product is nums[0] * nums[1] * ......

Practice › Codeforces Solutions › Codeforces Round 166 (Div. 2) ›

CF 271B - Prime Matrix

CF 271B - Prime Matrix Rating: 1300 Tags: binary search, brute force, math, number theory Solve time: 1m 46s Verified: yes Solution Problem Understanding We are given a rectangular grid of integers. In one operation we may pick any single cell and increase its value by exactly one. We can repeat this as many times as we want on any cells. The goal is not to make the whole matrix...

Practice › Codeforces Solutions › Codeforces Round 109 (Div. 1) ›

CF 154B - Colliders

CF 154B - Colliders Rating: 1600 Tags: math, number theory Solve time: 2m 4s Verified: yes Solution Problem Understanding We maintain a set of currently active colliders. Each collider is identified by an integer from 1 to n . The system is safe only if every pair of active colliders is coprime. In other words, no prime factor may appear in two different active numbers at the same time. Each...

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Codeforces Round 166 (Div. 2)

Codeforces Round 166 (Div. 2) Type: Div. 2 | Problems: 4 | Verified: 3/4 | Rating range: 1300-2200 | Time: 7m 9s Problem Name Rating Tags Solve Time Verified B Prime Matrix 1300 binary-search, brute-force, math 1m 46s ✓ C Secret 1500 constructive-algorithms, implementation 1m 31s ✗ D Good Substrings 1800 data-structures, strings 1m 42s ✓ E Three Horses 2200 constructive-algorithms, math, number-theory 2m 10s ✓

Practice › Codeforces Solutions › Croc Champ 2012 - Qualification Round ›

CF 172B - Pseudorandom Sequence Period

CF 172B - Pseudorandom Sequence Period Rating: 1200 Tags: *special, implementation, number theory Solve time: 3m 33s Verified: yes Solution Problem Understanding We are given a linear congruential generator, a classic pseudorandom sequence formula: $$r_i = (a \cdot r_{i-1} + b) \bmod m$$ The sequence starts from r0 , and every next value is computed from the previous one. Since every value is taken modulo m , there are only...

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LeetCode 2427 - Number of Common Factors

LeetCode Problem 2427 Difficulty: 🟢 Easy Topics: Math, Enumeration, Number Theory Solution Problem Understanding The problem gives us two positive integers, a and b , and asks us to count how many integers divide both numbers evenly. A number x is considered a common factor if: a % x == 0 b % x == 0 This means x divides both integers without leaving a remainder. For example, if a...

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Codeforces Round 240 (Div. 1)

Codeforces Round 240 (Div. 1) Type: Div. 1 | Problems: 5 | Verified: 2/5 | Rating range: 1400-3200 | Time: 8m 52s Problem Name Rating Tags Solve Time Verified A Mashmokh and Numbers 1500 constructive-algorithms, number-theory 2m 6s ✗ B Mashmokh and ACM 1400 combinatorics, dp, number-theory 1m 21s ✓ C Mashmokh and Reverse Operation 2100 combinatorics, divide-and-conquer 1m 40s ✓ D Mashmokh and Water Tanks 2300 binary-search, data-structures, greedy...

Practice › Codeforces Solutions › Codeforces Round 195 (Div. 2) ›

CF 336D - Vasily the Bear and Beautiful Strings

CF 336D - Vasily the Bear and Beautiful Strings Rating: 2100 Tags: combinatorics, math, number theory Solve time: 3m 11s Verified: yes Solution Problem Understanding We are asked to count the number of binary strings containing exactly n zeros and m ones that can be reduced, through a sequence of specific operations, to a single character g . The operation allowed is to take the last two characters of a...

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Codeforces Round 147 (Div. 2)

Codeforces Round 147 (Div. 2) Type: Div. 2 | Problems: 3 | Verified: 3/3 | Rating range: 1000-2000 | Time: 9m 35s Problem Name Rating Tags Solve Time Verified A Free Cash 1000 implementation 3m 17s ✓ C Primes on Interval 1600 binary-search, number-theory, two-pointers 4m 4s ✓ D T-decomposition 2000 dfs-and-similar, graphs, greedy 2m 14s ✓

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LeetCode 2654 - Minimum Number of Operations to Make All Array Elements Equal to 1

LeetCode Problem 2654 Difficulty: 🟡 Medium Topics: Array, Math, Number Theory Solution Problem Understanding The problem gives us an array of positive integers, nums . In one operation, we may choose two adjacent elements, nums[i] and nums[i+1] , and replace either one of them with the greatest common divisor, gcd , of the pair. The goal is to make every element in the array equal to 1 using the minimum...

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VK Cup 2012 Qualification Round 1

VK Cup 2012 Qualification Round 1 Type: Special | Problems: 5 | Verified: 5/5 | Rating range: 800-1900 | Time: 7m 26s Problem Name Rating Tags Solve Time Verified A Next Round 800 *special, implementation 1m 29s ✓ B Taxi 1100 *special, greedy, implementation 1m 21s ✓ C Cd and pwd commands 1400 *special, data-structures, implementation 1m 45s ✓ D Ice Sculptures 1300 *special, brute-force, number-theory 1m 38s ✓ E...

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LeetCode 2413 - Smallest Even Multiple

LeetCode Problem 2413 Difficulty: 🟢 Easy Topics: Math, Number Theory Solution Problem Understanding The problem asks us to find the smallest positive integer that is a multiple of both 2 and a given positive integer n . In other words, we are looking for the least common multiple (LCM) of 2 and n . The input n is guaranteed to be between 1 and 150 , which is a very...

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Codeforces Round 205 (Div. 2)

Codeforces Round 205 (Div. 2) Type: Div. 2 | Problems: 2 | Verified: 2/2 | Rating range: 1200-1600 | Time: 5m 32s Problem Name Rating Tags Solve Time Verified A Domino 1200 implementation, math 3m 55s ✓ C Find Maximum 1600 implementation, math, number-theory 1m 37s ✓

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LeetCode 3334 - Find the Maximum Factor Score of Array

LeetCode Problem 3334 Difficulty: 🟡 Medium Topics: Array, Math, Number Theory Solution Problem Understanding The problem asks us to compute the maximum factor score of an array of integers, where the factor score is defined as the product of the GCD (greatest common divisor) and LCM (least common multiple) of all elements in the array. We are allowed to remove at most one element to maximize this score. The input...

Practice › Codeforces Solutions › Codeforces Round 146 (Div. 2) ›

CF 236B - Easy Number Challenge

CF 236B - Easy Number Challenge Rating: 1300 Tags: implementation, number theory Solve time: 1m 45s Verified: yes Solution Problem Understanding We need to compute the sum of the number of divisors of every product i * j * k , where i ranges from 1 to a , j ranges from 1 to b , and k ranges from 1 to c . For each triple (i, j, k)...

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Testing Round 6

Testing Round 6 Type: Special | Problems: 4 | Verified: 1/4 | Rating range: 800-2300 | Time: 10m 15s Problem Name Rating Tags Solve Time Verified A Candies 800 implementation 3m 20s ✗ B Optimizer 2100 data-structures, greedy, sortings 1m 26s ✓ C White, Black and White Again 2100 combinatorics, number-theory 3m 20s ✗ D Polygon 2300 constructive-algorithms, geometry 2m 9s ✗

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LeetCode 2807 - Insert Greatest Common Divisors in Linked List

LeetCode Problem 2807 Difficulty: 🟡 Medium Topics: Linked List, Math, Number Theory Solution LeetCode 2807 - Insert Greatest Common Divisors in Linked List Problem Understanding This problem gives us the head of a singly linked list where every node contains a positive integer. Our task is to modify the list by inserting a new node between every pair of adjacent nodes. The value of each inserted node must be the...

Practice › Codeforces Solutions › Codeforces Round 205 (Div. 2) ›

CF 353C - Find Maximum

CF 353C - Find Maximum Rating: 1600 Tags: implementation, math, number theory Solve time: 1m 37s Verified: yes Solution Problem Understanding We are given an array a of n non-negative integers, and a number m specified as a binary string. Each integer in a represents a weight associated with a position, and for any number x between 0 and m inclusive, we can select positions where the bits of x...

Practice › Codeforces Solutions › Codeforces Round 206 (Div. 1) ›

CF 354C - Vasya and Beautiful Arrays

CF 354C - Vasya and Beautiful Arrays Rating: 2100 Tags: brute force, dp, number theory Solve time: 2m 47s Verified: yes Solution Problem Understanding We start with an array of positive integers. For every element, we are allowed to decrease it by at most k , and the value must stay positive. After performing these decreases independently on every position, we want the greatest common divisor of the resulting array...

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LeetCode 3348 - Smallest Divisible Digit Product II

LeetCode Problem 3348 Difficulty: 🔴 Hard Topics: Math, String, Backtracking, Greedy, Number Theory Solution Problem Understanding The problem asks us to find the smallest number that satisfies several constraints relative to a given number num and a target integer t . The input num is a string representing a positive integer, and the number must be zero-free , meaning none of its digits are zero. Additionally, the product of its...

Practice › Codeforces Solutions › Codeforces Round 153 (Div. 1) ›

CF 251C - Number Transformation

CF 251C - Number Transformation Rating: 2000 Tags: dp, greedy, number theory Solve time: 14m 23s Verified: no Solution

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LeetCode 3326 - Minimum Division Operations to Make Array Non Decreasing

LeetCode Problem 3326 Difficulty: 🟡 Medium Topics: Array, Math, Greedy, Number Theory Solution Problem Understanding The problem asks us to make a given integer array nums non-decreasing by performing a specific type of operation any number of times. Each operation consists of selecting an element and dividing it by its greatest proper divisor . A proper divisor of a number x is a positive integer less than x that divides...

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Codeforces Round 195 (Div. 2)

Codeforces Round 195 (Div. 2) Type: Div. 2 | Problems: 2 | Verified: 2/2 | Rating range: 2100-2700 | Time: 6m 33s Problem Name Rating Tags Solve Time Verified D Vasily the Bear and Beautiful Strings 2100 combinatorics, math, number-theory 3m 11s ✓ E Vasily the Bear and Painting Square 2700 bitmasks, combinatorics, dp 3m 22s ✓

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Codeforces Round 196 (Div. 1)

Codeforces Round 196 (Div. 1) Type: Div. 1 | Problems: 1 | Verified: 1/1 | Rating range: 2900-2900 | Time: 1m 33s Problem Name Rating Tags Solve Time Verified D GCD Table 2900 chinese-remainder-theorem, math, number-theory 1m 33s ✓

Practice › Codeforces Solutions › Codeforces Round 146 (Div. 1) ›

CF 235E - Number Challenge

CF 235E - Number Challenge Rating: 2600 Tags: combinatorics, dp, implementation, math, number theory Solve time: 3m 42s Verified: yes Solution Problem Understanding We need to compute $$\sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{k=1}^{c} d(i \cdot j \cdot k)$$ where $d(x)$ is the number of positive divisors of $x$. The three nested loops range over every triple $(i,j,k)$, multiply the numbers together, compute how many divisors that product has, and add everything modulo $2^{30} = 1073741824$....

Practice › Codeforces Solutions › Codeforces Round 133 (Div. 2) ›

CF 216E - Martian Luck

CF 216E - Martian Luck Rating: 2000 Tags: math, number theory Solve time: 1m 37s Verified: yes Solution Problem Understanding We are given a sequence of digits written in base k . Every digit is between 0 and k - 1 , and one special digit b is called lucky. For every substring of the digit sequence, we interpret that substring as a base- k number, allowing leading zeroes. A...

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Croc Champ 2012 - Qualification Round

Croc Champ 2012 - Qualification Round Type: Special | Problems: 5 | Verified: 4/5 | Rating range: 800-2200 | Time: 11m 42s Problem Name Rating Tags Solve Time Verified A Phone Code 800 *special, brute-force, implementation 3m ✓ B Pseudorandom Sequence Period 1200 *special, implementation, number-theory 3m 33s ✓ C Bus 1500 *special, implementation, sortings 1m 19s ✓ D Calendar Reform 1500 *special, number-theory 1m 29s ✓ E BHTML+BCSS 2200...

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LeetCode 2979 - Most Expensive Item That Can Not Be Bought

LeetCode Problem 2979 Difficulty: 🟡 Medium Topics: Math, Dynamic Programming, Number Theory Solution Problem Understanding This problem asks us to determine the largest price that cannot be formed using an unlimited number of coins of two given prime denominations. We are given two distinct prime numbers, primeOne and primeTwo . Alice has an infinite supply of coins with those exact denominations, and the market contains items of every positive integer...

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LeetCode 2464 - Minimum Subarrays in a Valid Split

LeetCode Problem 2464 Difficulty: 🟡 Medium Topics: Array, Math, Dynamic Programming, Number Theory Solution Problem Understanding The problem asks us to split an array of integers nums into contiguous subarrays such that each subarray satisfies a validity condition : the greatest common divisor (GCD) of the first and last elements of the subarray must be strictly greater than 1. Each element in nums must belong to exactly one subarray. Our...

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LeetCode 3312 - Sorted GCD Pair Queries

LeetCode Problem 3312 Difficulty: 🔴 Hard Topics: Array, Hash Table, Math, Binary Search, Combinatorics, Counting, Number Theory, Prefix Sum Solution === 1996-G1 === IMO 1996 Shortlist G1 Origin: GBR Problem Let triangle $ABC$ have orthocenter $H$, and let $P$ be a point on its circumcircle, distinct from $A$, $B$, $C$. Let $E$ be the foot of the altitude $BH$, let $PAQB$ and $PARC$ be parallelograms, and let $AQ$ meet $HR$...

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Codeforces Round 178 (Div. 2)

Codeforces Round 178 (Div. 2) Type: Div. 2 | Problems: 4 | Verified: 3/4 | Rating range: 800-2500 | Time: 6m 7s Problem Name Rating Tags Solve Time Verified A Shaass and Oskols 800 implementation, math 1m 38s ✓ C Shaass and Lights 1900 combinatorics, number-theory 1m 10s ✓ D Shaass and Painter Robot 2500 brute-force, implementation, number-theory 1m 47s ✗ E Shaass the Great 2300 dp, trees 1m 32s...

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LeetCode 2601 - Prime Subtraction Operation

LeetCode Problem 2601 Difficulty: 🟡 Medium Topics: Array, Math, Binary Search, Greedy, Number Theory Solution Problem Understanding The problem gives us an integer array nums and asks whether it is possible to transform the array into a strictly increasing sequence by performing a special operation. For any index i , we may choose that index at most once and subtract a prime number p from nums[i] , where p must...

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LeetCode 2344 - Minimum Deletions to Make Array Divisible

LeetCode Problem 2344 Difficulty: 🔴 Hard Topics: Array, Math, Sorting, Heap (Priority Queue), Number Theory Solution Problem Understanding The problem gives us two arrays of positive integers, nums and numsDivide . We are allowed to delete any number of elements from nums , and our goal is to make the smallest remaining element in nums divide every number in numsDivide . The key phrase here is smallest remaining element ....

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Codeforces Round 183 (Div. 1)

Codeforces Round 183 (Div. 1) Type: Div. 1 | Problems: 5 | Verified: 3/5 | Rating range: 1300-3000 | Time: 15m 59s Problem Name Rating Tags Solve Time Verified A Lucky Permutation Triple 1300 constructive-algorithms, implementation, math 2m 15s ✗ B Rectangle Puzzle II 1700 implementation, math 3m 45s ✓ C Minimum Modular 2400 brute-force, graphs, math 4m 56s ✓ D Rotatable Number 2600 math, number-theory 1m 56s ✓ E...

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LeetCode 2949 - Count Beautiful Substrings II

LeetCode Problem 2949 Difficulty: 🔴 Hard Topics: Hash Table, Math, String, Number Theory, Prefix Sum Solution Problem Understanding The problem asks us to find the number of non-empty substrings in a given string s that satisfy two conditions simultaneously. First, the substring must have an equal number of vowels and consonants. Second, the product of the number of vowels and consonants must be divisible by a given integer k ....

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CF 300E - Empire Strikes Back

CF 300E - Empire Strikes Back Rating: 2300 Tags: binary search, math, number theory Solve time: 1m 17s Verified: yes Solution Problem Understanding The problem asks us to determine the minimum positive integer $n$ such that the factorial of $n$, divided by the sum of the factorials of $n - a_i$ for a given sequence of integers $a_1, a_2, \dots, a_k$, results in a positive integer. Conceptually, we are looking...

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Codeforces Round #FF (Div. 1)

Codeforces Round #FF (Div. 1) Type: Div. 1 | Problems: 3 | Verified: 0/3 | Rating range: 2400-3100 | Time: 9m 38s Problem Name Rating Tags Solve Time Verified C DZY Loves Fibonacci Numbers 2400 data-structures, math, number-theory 6m 32s ✗ D DZY Loves Games 2800 math, matrices, probabilities 1m 41s ✗ E DZY Loves Bridges 3100 math, matrices 1m 25s ✗

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LeetCode 2818 - Apply Operations to Maximize Score

LeetCode Problem 2818 Difficulty: 🔴 Hard Topics: Array, Math, Stack, Greedy, Sorting, Monotonic Stack, Number Theory Solution Problem Understanding The problem gives us an array nums and an integer k . We begin with a score of 1 , and we are allowed to perform at most k operations. In each operation, we choose a subarray that has not been chosen before. From that subarray, we select the element with...

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LeetCode 2436 - Minimum Split Into Subarrays With GCD Greater Than One

LeetCode Problem 2436 Difficulty: 🟡 Medium Topics: Array, Math, Dynamic Programming, Greedy, Number Theory Solution Problem Understanding The problem requires splitting a given array of positive integers into contiguous subarrays such that the greatest common divisor (GCD) of each subarray is strictly greater than 1. The goal is to minimize the number of subarrays after splitting. In simpler terms, we need to partition the array into the fewest contiguous segments...

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CF 177B2 - Rectangular Game

CF 177B2 - Rectangular Game Rating: 1200 Tags: number theory Solve time: 2m 20s Verified: yes Solution Problem Understanding The problem describes a game played by a beaver with n pebbles. On each turn, the beaver arranges all remaining pebbles into a rectangle with more than one row and some number of columns. He then selects a single row and discards the rest. This repeats until only one pebble remains....

Practice › Codeforces Solutions › ABBYY Cup 2.0 - Easy ›

CF 177B1 - Rectangular Game

CF 177B1 - Rectangular Game Rating: 1000 Tags: number theory Solve time: 1m 36s Verified: yes Solution Problem Understanding The game starts with n pebbles. In one move, we split the current number of pebbles into a equal rows of size b , where a > 1 and a * b = current . After forming the rows, we keep exactly one row and throw away the others. That means...

Practice › Codeforces Solutions › Codeforces Round 235 (Div. 2) ›

CF 401D - Roman and Numbers

CF 401D - Roman and Numbers Rating: 2000 Tags: bitmasks, brute force, combinatorics, dp, number theory Solve time: 1m 19s Verified: yes Solution Problem Understanding We are asked to count the number of integers that can be formed by permuting the digits of a given number $n$, do not start with zero, and are divisible by a given modulus $m$. The input $n$ can be as large as $10^{18}$, which...

Practice › Codeforces Solutions › Codeforces Round 126 (Div. 2) ›

CF 200E - Tractor College

CF 200E - Tractor College Rating: 2400 Tags: implementation, math, number theory, ternary search Solve time: 1m 46s Verified: yes Solution Problem Understanding We are given the final exam results of a college where every student receives exactly one of three grades: 3, 4, or 5. Every student with the same grade must receive the same scholarship amount. If we denote these amounts as k3 , k4 , and k5...

Practice › LeetCode › LeetCode 29xx ›

LeetCode 2992 - Number of Self-Divisible Permutations

LeetCode Problem 2992 Difficulty: 🟡 Medium Topics: Array, Math, Dynamic Programming, Backtracking, Bit Manipulation, Number Theory, Bitmask Solution Problem Understanding The problem asks us to count how many permutations of the numbers 1 through n satisfy a special condition called self-divisible. We start with the array: [1, 2, 3, ..., n] We must rearrange these numbers into every possible permutation, then determine whether the permutation is valid. A permutation a...

Practice › Codeforces Solutions › Codeforces Round 139 (Div. 2) ›

CF 225B - Well-known Numbers

CF 225B - Well-known Numbers Rating: 1600 Tags: binary search, greedy, number theory Solve time: 3m 27s Verified: no Solution Problem Understanding We are given two integers, s and k . The task is to express s as a sum of distinct numbers taken from the k -bonacci sequence. The sequence behaves like Fibonacci, but instead of summing the previous two values, each term is the sum of the previous...

Practice › LeetCode › LeetCode 25xx ›

LeetCode 2507 - Smallest Value After Replacing With Sum of Prime Factors

LeetCode Problem 2507 Difficulty: 🟡 Medium Topics: Math, Simulation, Number Theory Solution Problem Understanding The problem asks us to repeatedly replace a number n with the sum of its prime factors until it reaches the smallest value it can take. Prime factors should be counted with multiplicity, meaning that if a prime factor divides n multiple times, it is added to the sum as many times as it divides n...

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LeetCode 2947 - Count Beautiful Substrings I

LeetCode Problem 2947 Difficulty: 🟡 Medium Topics: Hash Table, Math, String, Enumeration, Number Theory, Prefix Sum Solution LeetCode 2947 - Count Beautiful Substrings I Problem Understanding We are given a lowercase English string s and a positive integer k . We must count how many non-empty substrings satisfy the definition of a beautiful substring. For a substring, let: vowels be the number of vowel characters ( a , e ,...

Practice › Codeforces Solutions › Codeforces Round 132 (Div. 2) ›

CF 215E - Periodical Numbers

CF 215E - Periodical Numbers Rating: 2100 Tags: combinatorics, dp, number theory Solve time: 2m 58s Verified: no Solution Problem Understanding We are asked to count numbers in a given range [l, r] whose binary representations are periodic. A number is periodic if its binary string has a repeating pattern of length k that divides the total length n of the string, meaning every segment of length k repeats exactly...

Practice › Codeforces Solutions ›

Codeforces Round 146 (Div. 2)

Codeforces Round 146 (Div. 2) Type: Div. 2 | Problems: 2 | Verified: 2/2 | Rating range: 800-1300 | Time: 5m 12s Problem Name Rating Tags Solve Time Verified A Boy or Girl 800 brute-force, implementation, strings 3m 27s ✓ B Easy Number Challenge 1300 implementation, number-theory 1m 45s ✓

Practice › Codeforces Solutions ›

Croc Champ 2013 - Round 2

Croc Champ 2013 - Round 2 Type: Special | Problems: 4 | Verified: 4/4 | Rating range: 1500-2700 | Time: 8m 33s Problem Name Rating Tags Solve Time Verified A Weird Game 1500 games, greedy 53s ✓ C Cube Problem 2400 brute-force, math, number-theory 1m 36s ✓ D Ksusha and Square 2700 geometry, math, probabilities 1m 59s ✓ E Close Vertices 2700 data-structures, divide-and-conquer, trees 4m 5s ✓

Practice › LeetCode › LeetCode 29xx ›

LeetCode 2941 - Maximum GCD-Sum of a Subarray

LeetCode Problem 2941 Difficulty: 🔴 Hard Topics: Array, Math, Binary Search, Number Theory Solution Problem Understanding The problem asks us to find the maximum gcd-sum of a subarray of a given integer array nums with the constraint that the subarray has at least k elements. The gcd-sum of a subarray is defined as the product of the sum of its elements and the greatest common divisor (GCD) of its elements....

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LeetCode 3012 - Minimize Length of Array Using Operations

LeetCode Problem 3012 Difficulty: 🟡 Medium Topics: Array, Math, Greedy, Number Theory Solution Problem Understanding The problem gives us a 0-indexed array of positive integers and asks us to repeatedly perform a specific operation to reduce the array's length as much as possible. The operation involves selecting two distinct indices i and j with positive numbers, inserting nums[i] % nums[j] at the end, and removing the original two elements. The...

Practice › Codeforces Solutions › Codeforces Round 107 (Div. 1) ›

CF 150A - Win or Freeze

CF 150A - Win or Freeze Rating: 1400 Tags: games, math, number theory Solve time: 2m 2s Verified: no Solution Problem Understanding We start with a number q written on paper. On each turn, a player must replace the current number with one of its non-trivial divisors, meaning a divisor strictly between 1 and the number itself. If a player cannot make a move, that player wins immediately. This is...

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LeetCode 2709 - Greatest Common Divisor Traversal

LeetCode Problem 2709 Difficulty: 🔴 Hard Topics: Array, Math, Union-Find, Number Theory Solution Understood. I will provide the complete, detailed reference guide for LeetCode 2702 - Minimum Operations to Make Numbers Non-positive , following your formatting rules exactly. Problem Understanding The problem provides a 0-indexed integer array nums and two integers x and y . You are allowed to perform the following operation any number of times: choose an index...

Practice › Codeforces Solutions › Codeforces Round 142 (Div. 2) ›

CF 230B - T-primes

CF 230B - T-primes Rating: 1300 Tags: binary search, implementation, math, number theory Solve time: 55s Verified: yes Solution Problem Understanding We are asked to identify T-primes in a list of positive integers. A T-prime is defined as a number that has exactly three distinct positive divisors. Thinking about divisors, the only way a number can have exactly three is if it is the square of a prime number. For...

Practice › Codeforces Solutions › Codeforces Round 125 (Div. 2) ›

CF 199A - Hexadecimal's theorem

CF 199A - Hexadecimal's theorem Rating: 900 Tags: brute force, constructive algorithms, implementation, number theory Solve time: 1m 37s Verified: yes Solution Problem Understanding We are given a single integer n , and the statement guarantees that n itself is a Fibonacci number. The task is to represent n as the sum of three Fibonacci numbers. The three values are allowed to repeat, and zero is also a valid Fibonacci...

Practice › LeetCode › LeetCode 19xx ›

LeetCode 1952 - Three Divisors

LeetCode Problem 1952 Difficulty: 🟢 Easy Topics: Math, Enumeration, Number Theory Solution Problem Understanding The problem asks us to determine whether a given integer n has exactly three positive divisors . In other words, we want to check if there are precisely three distinct integers d such that n % d == 0 and d > 0 . For example, the number 4 has divisors 1, 2, and 4, which...

Practice › Codeforces Solutions › Codeforces Round 166 (Div. 2) ›

CF 271E - Three Horses

CF 271E - Three Horses Rating: 2200 Tags: constructive algorithms, math, number theory Solve time: 2m 10s Verified: yes Solution Problem Understanding We start with exactly one card (x, y) where 1 ≤ x < y ≤ m . Three operations are available. The gray horse increases both numbers by one: (a, b) → (a + 1, b + 1) . The white horse works only when both numbers are...

Practice › Codeforces Solutions ›

Codeforces Round 256 (Div. 2)

Codeforces Round 256 (Div. 2) Type: Div. 2 | Problems: 3 | Verified: 2/3 | Rating range: 800-2200 | Time: 3m 43s Problem Name Rating Tags Solve Time Verified A Rewards 800 implementation 1m 14s ✓ D Multiplication Table 1800 binary-search, brute-force 1m 14s ✓ E Divisors 2200 brute-force, dfs-and-similar, implementation 1m 15s ✗

Practice › Codeforces Solutions › Codeforces Testing Round 5 ›

CF 267A - Subtractions

CF 267A - Subtractions Rating: 900 Tags: math, number theory Solve time: 2m 41s Verified: yes Solution Problem Understanding We repeatedly apply the same operation to two positive integers. At every step, we subtract the smaller value from the larger one. The process stops as soon as one number becomes zero. The task is to count how many subtraction operations are performed for each pair. For example, starting from (4,...

Practice › LeetCode › LeetCode 30xx ›

LeetCode 3044 - Most Frequent Prime

LeetCode Problem 3044 Difficulty: 🟡 Medium Topics: Array, Hash Table, Math, Matrix, Counting, Enumeration, Number Theory Solution Problem Understanding The problem asks us to traverse a m x n integer matrix and generate numbers by moving in straight lines along eight possible directions: east, south-east, south, south-west, west, north-west, north, and north-east. Starting from any cell, we can append the digits along a chosen path to form numbers. Importantly, numbers...

Practice › Codeforces Solutions ›

Codeforces Round 126 (Div. 2)

Codeforces Round 126 (Div. 2) Type: Div. 2 | Problems: 3 | Verified: 3/3 | Rating range: 800-2400 | Time: 7m 2s Problem Name Rating Tags Solve Time Verified B Drinks 800 implementation, math 1m 33s ✓ C Football Championship 1800 brute-force, implementation 3m 43s ✓ E Tractor College 2400 implementation, math, number-theory 1m 46s ✓

Practice › LeetCode › LeetCode 24xx ›

LeetCode 2447 - Number of Subarrays With GCD Equal to K

LeetCode Problem 2447 Difficulty: 🟡 Medium Topics: Array, Math, Number Theory Solution LeetCode 2447 - Number of Subarrays With GCD Equal to K Problem Understanding The problem gives us an integer array nums and an integer k . We need to count how many contiguous subarrays have a greatest common divisor (GCD) exactly equal to k . A subarray is any non-empty contiguous segment of the array. For every possible...

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LeetCode 2470 - Number of Subarrays With LCM Equal to K

LeetCode Problem 2470 Difficulty: 🟡 Medium Topics: Array, Math, Number Theory Solution LeetCode 2470, Number of Subarrays With LCM Equal to K Problem Understanding The problem gives us an integer array nums and an integer k . We must count how many contiguous subarrays have a least common multiple, LCM, exactly equal to k . A subarray is any continuous segment of the array. For every possible subarray, we compute...

Practice › LeetCode › LeetCode 28xx ›

LeetCode 2862 - Maximum Element-Sum of a Complete Subset of Indices

LeetCode Problem 2862 Difficulty: 🔴 Hard Topics: Array, Math, Number Theory Solution LeetCode 2862 - Maximum Element-Sum of a Complete Subset of Indices Problem Understanding The array nums is 1-indexed , meaning the first element corresponds to index 1 , the second element corresponds to index 2 , and so on. We want to select a subset of indices such that for every pair of selected indices i and j...

Practice › LeetCode › LeetCode 25xx ›

LeetCode 2572 - Count the Number of Square-Free Subsets

LeetCode Problem 2572 Difficulty: 🟡 Medium Topics: Array, Math, Dynamic Programming, Bit Manipulation, Number Theory, Bitmask Solution Problem Understanding The problem asks us to count all non-empty subsets of a given array nums such that the product of the elements in each subset is square-free . A square-free integer is an integer not divisible by the square of any prime greater than 1. In other words, in its prime factorization,...

Practice › Codeforces Solutions › Codeforces Round 139 (Div. 2) ›

CF 225E - Unsolvable

CF 225E - Unsolvable Rating: 2100 Tags: math, number theory Solve time: 3m 28s Verified: yes Solution Problem Understanding The equation in the statement is $$\left\lfloor \frac{x}{y} \right\rfloor + \left\lfloor \frac{y}{x} \right\rfloor = z$$ and we want all positive integers $z$ for which this equation has no solution in positive integers $x,y$. The sequence of such numbers is increasing: $$z_1, z_2, z_3, \dots$$ Given $n$, we must output $z_n \bmod...

Practice › LeetCode › LeetCode 21xx ›

LeetCode 2197 - Replace Non-Coprime Numbers in Array

LeetCode Problem 2197 Difficulty: 🔴 Hard Topics: Array, Math, Stack, Number Theory Solution Problem Understanding The problem asks us to repeatedly replace adjacent non-coprime numbers in an array with their Least Common Multiple (LCM) until no more adjacent non-coprime pairs exist. Two numbers are non-coprime if their greatest common divisor (GCD) is greater than 1. The final array should be returned after all possible replacements. In other words, given an...

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LeetCode 3115 - Maximum Prime Difference

LeetCode Problem 3115 Difficulty: 🟡 Medium Topics: Array, Math, Number Theory Solution Problem Understanding The problem gives us an integer array nums , and we need to find the maximum distance between the indices of any two prime numbers in the array. More specifically, we are interested in indices i and j such that both nums[i] and nums[j] are prime numbers. Among all such pairs, we want the maximum possible...

Practice › Codeforces Solutions › Codeforces Round #FF (Div. 1) ›

CF 446C - DZY Loves Fibonacci Numbers

CF 446C - DZY Loves Fibonacci Numbers Rating: 2400 Tags: data structures, math, number theory Solve time: 6m 32s Verified: no Solution Problem Understanding We are given an array of integers and need to support two types of queries efficiently. The first type requires adding Fibonacci numbers to a contiguous subarray: for indices from l to r , we add F₁ to the element at l , F₂ to the...

Practice › Codeforces Solutions › Codeforces Round 201 (Div. 1) ›

CF 346A - Alice and Bob

CF 346A - Alice and Bob Rating: 1600 Tags: games, math, number theory Solve time: 3m 49s Verified: yes Solution Problem Understanding We start with a collection of distinct integers. On each move, the current player picks any two numbers from the set and inserts their absolute difference, but only if that difference is not already present. The set grows over time, and the game ends when a player cannot...

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Codeforces Round 109 (Div. 1)

Codeforces Round 109 (Div. 1) Type: Div. 1 | Problems: 4 | Verified: 4/4 | Rating range: 1600-2400 | Time: 6m 30s Problem Name Rating Tags Solve Time Verified A Hometask 1600 greedy 1m 15s ✓ B Colliders 1600 math, number-theory 2m 4s ✓ C Double Profiles 2300 graphs, hashing, sortings 1m 37s ✓ D Flatland Fencing 2400 games, math 1m 34s ✓

Practice › Codeforces Solutions › Codeforces Round 140 (Div. 1) ›

CF 226C - Anniversary

CF 226C - Anniversary Rating: 2400 Tags: data structures, implementation, math, matrices, number theory Solve time: 2m 19s Verified: yes Solution Problem Understanding We are asked to work with a contiguous range of integers from l to r , and for every possible subset of size k within this range, we consider the Fibonacci numbers at positions given by the subset elements. Among all subsets, we must find the maximum...

Practice › Codeforces Solutions ›

ABBYY Cup 2.0 - Easy

ABBYY Cup 2.0 - Easy Type: Special | Problems: 11 | Verified: 10/11 | Rating range: 800-2600 | Time: 24m 43s Problem Name Rating Tags Solve Time Verified A1 Good Matrix Elements 800 implementation 1m 37s ✗ A2 Good Matrix Elements 800 implementation 3m 13s ✓ B1 Rectangular Game 1000 number-theory 1m 36s ✓ B2 Rectangular Game 1200 number-theory 2m 20s ✓ C1 Party 1500 dfs-and-similar, dsu, graphs 3m 2s...

Practice › Codeforces Solutions › Codeforces Round 240 (Div. 1) ›

CF 414B - Mashmokh and ACM

CF 414B - Mashmokh and ACM Rating: 1400 Tags: combinatorics, dp, number theory Solve time: 1m 21s Verified: yes Solution Problem Understanding We are asked to count how many sequences of fixed length we can build from integers between 1 and n, with two constraints. First, the sequence is non-decreasing. Second, every element must divide the next one in the sequence. So if we pick a sequence like b1, b2,...

Practice › Codeforces Solutions › Codeforces Round 162 (Div. 1) ›

CF 264B - Good Sequences

CF 264B - Good Sequences Rating: 1500 Tags: dp, number theory Solve time: 1m 56s Verified: yes Solution Problem Understanding We are given a strictly increasing array of integers. We want to build the longest subsequence such that every pair of neighboring numbers shares at least one common prime factor. Since the original array is already strictly increasing, any subsequence automatically preserves increasing order. The real condition is about adjacency:...

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Codeforces Round 236 (Div. 2)

Codeforces Round 236 (Div. 2) Type: Div. 2 | Problems: 4 | Verified: 1/4 | Rating range: 1100-2200 | Time: 11m 10s Problem Name Rating Tags Solve Time Verified A Nuts 1100 greedy, math 4m 11s ✗ B Trees in a Row 1400 brute-force, implementation 2m 19s ✗ D Upgrading Array 1800 dp, greedy, math 1m 3s ✓ E Strictly Positive Matrix 2200 graphs, math 3m 37s ✗

Practice › LeetCode › LeetCode 27xx ›

LeetCode 2761 - Prime Pairs With Target Sum

LeetCode Problem 2761 Difficulty: 🟡 Medium Topics: Array, Math, Enumeration, Number Theory Solution Problem Understanding The problem asks us to find all pairs of prime numbers (x, y) such that both numbers are between 1 and n inclusive, their sum equals n , and x <= y . The input is a single integer n , and the output is a sorted list of pairs [x, y] meeting these conditions,...

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LeetCode 2521 - Distinct Prime Factors of Product of Array

LeetCode Problem 2521 Difficulty: 🟡 Medium Topics: Array, Hash Table, Math, Number Theory Solution Problem Understanding The problem asks us to compute the number of distinct prime factors in the product of an array of positive integers, nums . In simpler terms, we are given a list of numbers, and we want to know how many unique prime numbers appear if we were to multiply all the numbers together and...

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LeetCode 3377 - Digit Operations to Make Two Integers Equal

LeetCode Problem 3377 Difficulty: 🟡 Medium Topics: Math, Graph Theory, Heap (Priority Queue), Number Theory, Shortest Path Solution Problem Understanding The problem asks us to transform one integer n into another integer m by repeatedly modifying individual digits of n . At each step, we can increase a digit that is not 9 or decrease a digit that is not 0. However, there is a strict constraint: n cannot ever...

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LeetCode 1998 - GCD Sort of an Array

LeetCode Problem 1998 Difficulty: 🔴 Hard Topics: Array, Math, Union-Find, Sorting, Number Theory Solution Problem Understanding The problem asks whether it is possible to sort an integer array nums into non-decreasing order using a very specific type of swap. You can swap any two elements nums[i] and nums[j] if and only if their greatest common divisor (GCD) is greater than 1. The input is an array of integers ranging from...

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LeetCode 3233 - Find the Count of Numbers Which Are Not Special

LeetCode Problem 3233 Difficulty: 🟡 Medium Topics: Array, Math, Number Theory Solution Problem Understanding The problem asks us to count how many integers in the inclusive range [l, r] are not special . A number is considered special if it has exactly two proper divisors. A proper divisor of a number x is any positive divisor of x other than x itself. For example: Proper divisors of 4 are 1...

Practice › Codeforces Solutions › Codeforces Round 116 (Div. 2, ACM-ICPC Rules) ›

CF 180B - Divisibility Rules

CF 180B - Divisibility Rules Rating: 2300 Tags: math, number theory Solve time: 1m 50s Verified: yes Solution Problem Understanding We work in a positional numeral system with base b . For a divisor d , we must determine which kind of divisibility rule exists in this base. The problem defines several categories. A divisor belongs to the 2-type category if divisibility depends only on a fixed number of last...

Practice › Codeforces Solutions › Codeforces Round 196 (Div. 1) ›

CF 338D - GCD Table

CF 338D - GCD Table Rating: 2900 Tags: chinese remainder theorem, math, number theory Solve time: 1m 33s Verified: yes Solution Problem Understanding We are asked to determine whether a given sequence of numbers appears as consecutive elements in some row of a hypothetical GCD table. The table has n rows and m columns, where each element at row i and column j is the greatest common divisor of i...

Practice › Codeforces Solutions › Codeforces Testing Round 3 ›

CF 134B - Pairs of Numbers

CF 134B - Pairs of Numbers Rating: 1900 Tags: brute force, dfs and similar, math, number theory Solve time: 1m 30s Verified: yes Solution Problem Understanding We are asked to start with the number pair (1,1) and reach a pair where at least one of the numbers equals a given target n . At each step, we can add one number to the other to form a new pair. Concretely,...

Practice › Codeforces Solutions › Codeforces Beta Round 88 ›

CF 117B - Very Interesting Game

CF 117B - Very Interesting Game Rating: 1800 Tags: brute force, number theory Solve time: 2m 31s Verified: yes Solution Problem Understanding We are asked to simulate a two-player game involving numbers represented as nine-digit strings. The first player selects a string s1 that represents a number not exceeding a given value a . The second player responds with a string s2 representing a number not exceeding b . Both...

Practice › Codeforces Solutions › Codeforces Beta Round 65 (Div. 2) ›

CF 71C - Round Table Knights

CF 71C - Round Table Knights Rating: 1600 Tags: dp, math, number theory Solve time: 1m 40s Verified: yes Solution Problem Understanding The knights sit on a circle, equally spaced. Each position is marked either 1 for a knight in a good mood or 0 for a knight in a bad mood. We need to determine whether it is possible to choose some of the good-mood knights so that they...

Practice › LeetCode › LeetCode 02xx ›

LeetCode 258 - Add Digits

LeetCode Problem 258 Difficulty: 🟢 Easy Topics: Math, Simulation, Number Theory Solution Problem Understanding The problem asks us to repeatedly transform a number by summing its digits until only a single digit remains. The final single digit should then be returned. For example, if the input is 38 , we first compute 3 + 8 = 11 . Since 11 still has more than one digit, we repeat the process...

Practice › LeetCode › LeetCode 16xx ›

LeetCode 1627 - Graph Connectivity With Threshold

LeetCode Problem 1627 Difficulty: 🔴 Hard Topics: Array, Math, Union-Find, Number Theory Solution Problem Understanding The problem gives us n cities labeled from 1 to n . Two cities are directly connected if they share a common divisor that is strictly greater than a given threshold . For example, if threshold = 2 , then cities 6 and 9 are directly connected because they share divisor 3 , and 3...

Practice › LeetCode › LeetCode 17xx ›

LeetCode 1766 - Tree of Coprimes

LeetCode Problem 1766 Difficulty: 🔴 Hard Topics: Array, Math, Tree, Depth-First Search, Number Theory Solution Problem Understanding This problem gives us a tree with n nodes numbered from 0 to n-1 , rooted at node 0 . Each node has an associated value from the array nums , and the tree structure is given as a list of edges. For each node i , we are asked to find the...

Practice › Codeforces Solutions › Codeforces Beta Round 55 (Div. 2) ›

CF 59B - Fortune Telling

CF 59B - Fortune Telling Rating: 1200 Tags: implementation, number theory Solve time: 1m 57s Verified: no Solution Problem Understanding Marina can pick any subset of flowers from the field. Each flower has a certain number of petals, and she will pluck all petals from all chosen flowers one by one. The phrases alternate between "Loves" and "Doesn't love" , starting from "Loves" on the first petal. The final phrase...

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Codeforces Beta Round 91 (Div. 2 Only)

Codeforces Beta Round 91 (Div. 2 Only) Type: Div. 2 | Problems: 2 | Verified: 2/2 | Rating range: 1000-1000 | Time: 2m 30s Problem Name Rating Tags Solve Time Verified A Lucky Division 1000 brute-force, number-theory 1m 12s ✓ B Lucky Substring 1000 brute-force, implementation 1m 18s ✓

Practice › LeetCode › LeetCode 09xx ›

LeetCode 952 - Largest Component Size by Common Factor

LeetCode Problem 952 Difficulty: 🔴 Hard Topics: Array, Hash Table, Math, Union-Find, Number Theory Solution Problem Understanding The problem asks us to find the largest connected component in a graph derived from an array of unique positive integers nums . Each integer in the array represents a node, and there is an undirected edge between two nodes if the corresponding numbers share a common factor greater than 1. Essentially, numbers...

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Codeforces Beta Round 62

Codeforces Beta Round 62 Type: Beta | Problems: 5 | Verified: 5/5 | Rating range: 1100-2900 | Time: 26m 18s Problem Name Rating Tags Solve Time Verified A Irrational problem 1100 implementation, number-theory 7m 4s ✓ B Energy exchange 1600 binary-search 7m 4s ✓ C Synchrophasotron 2200 brute-force 7m 3s ✓ D Half-decay tree 2500 data-structures, divide-and-conquer, dp 2m 50s ✓ E Contact 2900 geometry 2m 17s ✓

Practice › Codeforces Solutions › School Team Contest 3 (Winter Computer School 2010/11) ›

CF 45G - Prime Problem

CF 45G - Prime Problem Rating: 2200 Tags: number theory Solve time: 2m 16s Verified: no Solution Problem Understanding We have houses numbered from 1 to n . Every house must receive a color. For each color, if we add together all house indices painted with that color, the result must be a prime number. The goal is not to maximize anything about the partition itself. The only optimization target...

Practice › Codeforces Solutions › Codeforces Beta Round 86 (Div. 1 Only) ›

CF 113C - Double Happiness

CF 113C - Double Happiness Rating: 2200 Tags: brute force, math, number theory Solve time: 3m 20s Verified: yes Solution Problem Understanding We are asked to count the numbers in a given interval $l, r$ that are simultaneously prime and expressible as the sum of two positive squares. In simpler terms, for each number in the interval, we need to check two independent properties. First, whether it is prime -...

Practice › Codeforces Solutions › Codeforces Beta Round 91 (Div. 2 Only) ›

CF 122A - Lucky Division

CF 122A - Lucky Division Rating: 1000 Tags: brute force, number theory Solve time: 1m 12s Verified: yes Solution Problem Understanding We are asked to determine whether a given number is almost lucky. A number is almost lucky if it is divisible by at least one lucky number. Lucky numbers are positive integers composed entirely of the digits 4 and 7, like 4, 7, 44, 47, 74, and so on....

Practice › Codeforces Solutions › Codeforces Beta Round 51 ›

CF 55D - Beautiful numbers

CF 55D - Beautiful numbers Rating: 2500 Tags: dp, number theory Solve time: 3m 29s Verified: no Solution Problem Understanding We are asked to count numbers in given ranges that Volodya would call beautiful. A number is beautiful if it is divisible by each of its nonzero digits. For example, 128 is beautiful because 128 is divisible by 1, 2, and 8. A single test case gives a lower bound...

Practice › Codeforces Solutions › Codeforces Beta Round 92 (Div. 1 Only) ›

CF 123A - Prime Permutation

CF 123A - Prime Permutation Rating: 1300 Tags: implementation, number theory, strings Solve time: 2m 19s Verified: no Solution Problem Understanding We are given a string s composed of lowercase letters, with length n . The task is to determine if we can rearrange the letters so that for every prime p less than or equal to n , all positions in the string that are multiples of p can...

Practice › LeetCode › LeetCode 16xx ›

LeetCode 1622 - Fancy Sequence

LeetCode Problem 1622 Difficulty: 🔴 Hard Topics: Math, Design, Segment Tree, Number Theory Solution Problem Understanding The problem asks us to design a mutable sequence data structure that supports four operations efficiently: Append a value to the sequence. Add a constant value to every element currently in the sequence. Multiply every element currently in the sequence by a constant. Retrieve the value at a specific index. All returned values must...

Practice › LeetCode › LeetCode 12xx ›

LeetCode 1201 - Ugly Number III

LeetCode Problem 1201 Difficulty: 🟡 Medium Topics: Math, Binary Search, Combinatorics, Number Theory Solution Problem Understanding The problem defines an "ugly number" as any positive integer that is divisible by at least one of the three given integers a , b , or c . We are given four integers: n , representing which ugly number we want to find a , b , and c , representing the divisibility...

Practice › Codeforces Solutions › Codeforces Beta Round 85 (Div. 1 Only) ›

CF 111B - Petya and Divisors

CF 111B - Petya and Divisors Rating: 1900 Tags: binary search, data structures, number theory Solve time: 2m 50s Verified: yes Solution Problem Understanding We are asked to process a sequence of queries, each defined by two integers, x and y. For each query, we need to count how many positive divisors of x do not divide any of the y numbers immediately preceding x in the sequence, that is,...

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LeetCode 1735 - Count Ways to Make Array With Product

LeetCode Problem 1735 Difficulty: 🔴 Hard Topics: Array, Math, Dynamic Programming, Combinatorics, Number Theory Solution Problem Understanding The problem asks us to count the number of ways to fill an array of size ni with positive integers such that the product of all elements equals ki . Each query in the input array queries is independent, meaning we compute the answer for each (ni, ki) pair separately. The output is...

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Codeforces Beta Round 16 (Div. 2 Only)

Codeforces Beta Round 16 (Div. 2 Only) Type: Div. 2 | Problems: 5 | Verified: 5/5 | Rating range: 800-1900 | Time: 7m 6s Problem Name Rating Tags Solve Time Verified A Flag 800 implementation 1m 8s ✓ B Burglar and Matches 900 greedy, implementation, sortings 1m 13s ✓ C Monitor 1800 binary-search, number-theory 1m 37s ✓ D Logging 1900 implementation, strings 1m 34s ✓ E Fish 1900 bitmasks, dp,...

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LeetCode 858 - Mirror Reflection

LeetCode Problem 858 Difficulty: 🟡 Medium Topics: Math, Geometry, Number Theory Solution LeetCode 858 - Mirror Reflection Problem Understanding This problem describes a square room with perfectly reflective walls. The room has side length p , and there are three receptors placed at three corners of the square: Receptor 0 is at the southeast corner Receptor 1 is at the northeast corner Receptor 2 is at the northwest corner The...

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CF 26A - Almost Prime

CF 26A - Almost Prime Rating: 900 Tags: number theory Solve time: 1m 18s Verified: yes Solution Problem Understanding We are asked to count numbers between 1 and $n$ that have exactly two distinct prime factors. For instance, 6 is almost prime because it can be factored as $2 \cdot 3$, and both 2 and 3 are prime. Numbers like 4 or 8 are not almost prime because they are...

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CF 27E - Number With The Given Amount Of Divisors

CF 27E - Number With The Given Amount Of Divisors Rating: 2000 Tags: brute force, dp, number theory Solve time: 1m 34s Verified: yes Solution Problem Understanding We are given a single integer n , and we need to construct the smallest positive integer whose number of divisors is exactly n . The core mathematical fact behind the problem is the divisor formula. If a number has prime factorization $$x...

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CF 16C - Monitor

CF 16C - Monitor Rating: 1800 Tags: binary search, number theory Solve time: 1m 37s Verified: yes Solution Problem Understanding We start with a monitor whose dimensions are a × b . Both values are integers. We want to shrink this monitor so that the new dimensions keep the exact aspect ratio x : y . The new monitor must satisfy three conditions at the same time. First, both dimensions...

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Codeforces Beta Round 85 (Div. 1 Only)

Codeforces Beta Round 85 (Div. 1 Only) Type: Div. 1 | Problems: 5 | Verified: 3/5 | Rating range: 1400-2900 | Time: 12m 59s Problem Name Rating Tags Solve Time Verified A Petya and Inequiations 1400 greedy 2m 33s ✗ B Petya and Divisors 1900 binary-search, data-structures, number-theory 2m 50s ✓ C Petya and Spiders 2100 bitmasks, dp, dsu 2m 34s ✓ D Petya and Coloring 2300 combinatorics, dp 2m...

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Codeforces Beta Round 92 (Div. 1 Only)

Codeforces Beta Round 92 (Div. 1 Only) Type: Div. 1 | Problems: 5 | Verified: 3/5 | Rating range: 1300-2500 | Time: 14m 53s Problem Name Rating Tags Solve Time Verified A Prime Permutation 1300 implementation, number-theory, strings 2m 19s ✗ B Squares 1800 math 5m 4s ✓ C Brackets 2300 combinatorics, dp, greedy 2m 19s ✓ D String 2300 string-suffix-structures 2m 32s ✓ E Maze 2500 dfs-and-similar, dp, probabilities...

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CF 68A - Irrational problem

CF 68A - Irrational problem Rating: 1100 Tags: implementation, number theory Solve time: 7m 4s Verified: yes Solution Problem Understanding We are asked to count integers $x$ in a given range $[a, b]$ that satisfy a certain remainder-based property. Petya has four distinct integers $p_1, p_2, p_3, p_4$. For any $x$, he can repeatedly take remainders of $x$ modulo these four numbers, but the order of operations is unknown. That...

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CF 73E - Morrowindows

CF 73E - Morrowindows Rating: 2400 Tags: math, number theory Solve time: 1m 51s Verified: yes Solution Problem Understanding Each viewing mode groups the inventory into pages of size a_i . If the inventory contains k items, then the game shows $b_i = \left\lceil \frac{k}{a_i} \right\rceil$ pages in that mode. Vasya does not know the actual value of k , only that 2 ≤ k ≤ x . Before looking...

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Codeforces Beta Round 67 (Div. 2)

Codeforces Beta Round 67 (Div. 2) Type: Div. 2 | Problems: 5 | Verified: 4/5 | Rating range: 1000-2400 | Time: 9m 31s Problem Name Rating Tags Solve Time Verified A Life Without Zeros 1000 implementation 1m 34s ✓ B Facetook Priority Wall 1500 expression-parsing, implementation, strings 2m 14s ✗ C Modified GCD 1600 binary-search, number-theory 1m 40s ✓ D Big Maximum Sum 2000 data-structures, dp, greedy 1m 34s ✓...

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CF 74C - Chessboard Billiard

CF 74C - Chessboard Billiard Rating: 2100 Tags: dfs and similar, dsu, graphs, number theory Solve time: 1m 36s Verified: yes Solution Problem Understanding We have an n × m chessboard. A billiard ball moves diagonally like a bishop, but unlike a bishop it reflects off the borders of the board. When it hits a vertical wall, the horizontal component of the direction flips. When it hits a horizontal wall,...

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LeetCode 914 - X of a Kind in a Deck of Cards

LeetCode Problem 914 Difficulty: 🟢 Easy Topics: Array, Hash Table, Math, Counting, Number Theory Solution Problem Understanding The problem asks whether it is possible to partition a deck of cards, represented as an integer array deck , into groups such that each group contains exactly x cards, all of which have the same integer value, and x is greater than 1. Essentially, we need to find a number x that...

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Codeforces Testing Round 3

Codeforces Testing Round 3 Type: Special | Problems: 3 | Verified: 2/3 | Rating range: 1200-2200 | Time: 5m 24s Problem Name Rating Tags Solve Time Verified A Average Numbers 1200 brute-force, implementation 1m 39s ✓ B Pairs of Numbers 1900 brute-force, dfs-and-similar, math 1m 30s ✓ C Swaps 2200 constructive-algorithms, graphs, greedy 2m 15s ✗

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CF 75C - Modified GCD

CF 75C - Modified GCD Rating: 1600 Tags: binary search, number theory Solve time: 1m 40s Verified: yes Solution Problem Understanding We are given two positive integers, and then many range queries. For each query [low, high] , we must find the largest integer inside that interval that divides both numbers. The first observation is that we are not really working with a and b independently. Any number that divides...

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CF 83D - Numbers

CF 83D - Numbers Rating: 2400 Tags: dp, math, number theory Solve time: 2m 7s Verified: yes Solution Problem Understanding We need to count integers inside the interval $[a,b]$ whose smallest divisor greater than $1$ is exactly $k$. For a number $x$, saying that its smallest divisor is $k$ means two things happen simultaneously: $x$ is divisible by $k$. $x$ is not divisible by any integer from $2$ to $k-1$....

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Codeforces Beta Round 7

Codeforces Beta Round 7 Type: Beta | Problems: 5 | Verified: 5/5 | Rating range: 1100-2600 | Time: 5m 10s Problem Name Rating Tags Solve Time Verified A Kalevitch and Chess 1100 brute-force, constructive-algorithms 1m 5s ✓ B Memory Manager 1600 implementation 1m 1s ✓ C Line 1800 math, number-theory 52s ✓ D Palindrome Degree 2200 hashing, strings 1m 5s ✓ E Defining Macros 2600 dp, expression-parsing, implementation 1m 7s...

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LeetCode 1447 - Simplified Fractions

LeetCode Problem 1447 Difficulty: 🟡 Medium Topics: Math, String, Number Theory Solution Problem Understanding The problem asks us to generate all simplified fractions between 0 and 1 (exclusive) where the denominator does not exceed a given integer n . A simplified fraction is one in which the numerator and denominator are coprime , meaning their greatest common divisor (GCD) is 1. For example, 2/4 is not simplified because it reduces...

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Codeforces Beta Round 70 (Div. 2)

Codeforces Beta Round 70 (Div. 2) Type: Div. 2 | Problems: 5 | Verified: 4/5 | Rating range: 800-2300 | Time: 12m 16s Problem Name Rating Tags Solve Time Verified A Haiku 800 implementation, strings 3m 30s ✓ B Easter Eggs 1200 constructive-algorithms, implementation 1m 59s ✗ C Beaver Game 2000 dp, games, number-theory 2m 6s ✓ D Archer's Shot 2300 binary-search, geometry, math 2m 7s ✓ E Evacuation 2300...

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CF 121C - Lucky Permutation

CF 121C - Lucky Permutation Rating: 1900 Tags: brute force, combinatorics, number theory Solve time: 1m 34s Verified: yes Solution Problem Understanding We are asked to examine permutations of numbers from 1 to n and focus on "lucky numbers" - integers that contain only the digits 4 and 7. The problem specifically asks us to find a lexicographically k -th permutation of numbers from 1 to n , and count...

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CF 17A - Noldbach problem

CF 17A - Noldbach problem Rating: 1000 Tags: brute force, math, number theory Solve time: 1m 46s Verified: yes Solution Problem Understanding We are given two integers, n and k . We look at all prime numbers from 2 up to n . Among those primes, we want to count how many can be written in the form: $$p_i + p_{i+1} + 1$$ where p_i and p_{i+1} are neighboring primes,...

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Codeforces Beta Round 10

Codeforces Beta Round 10 Type: Beta | Problems: 5 | Verified: 4/5 | Rating range: 900-2800 | Time: 8m 3s Problem Name Rating Tags Solve Time Verified A Power Consumption Calculation 900 implementation 1m 30s ✓ B Cinema Cashier 1500 dp, implementation 1m 24s ✓ C Digital Root 2000 number-theory 1m 41s ✓ D LCIS 2800 dp 2m ✗ E Greedy Change 2600 constructive-algorithms 1m 28s ✓

Practice › Codeforces Solutions › Codeforces Beta Round 72 (Div. 2 Only) ›

CF 84A - Toy Army

CF 84A - Toy Army Rating: 900 Tags: math, number theory Solve time: 1m 30s Verified: yes Solution Problem Understanding We have two armies, each containing n soldiers. The value of n is always even. The game lasts exactly three turns: Valera attacks Arcady. Arcady attacks Valera. Valera attacks Arcady again. Every surviving soldier from the current player chooses exactly one target from the opposing army. All shots happen simultaneously,...

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Codeforces Beta Round 55 (Div. 2)

Codeforces Beta Round 55 (Div. 2) Type: Div. 2 | Problems: 5 | Verified: 3/5 | Rating range: 800-2000 | Time: 8m 58s Problem Name Rating Tags Solve Time Verified A Word 800 implementation, strings 1m 11s ✓ B Fortune Telling 1200 implementation, number-theory 1m 57s ✗ C Title 1600 expression-parsing 1m 53s ✓ D Team Arrangement 2000 constructive-algorithms, greedy, implementation 2m 39s ✗ E Shortest Path 2000 graphs, shortest-paths...

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LeetCode 1492 - The kth Factor of n

LeetCode Problem 1492 Difficulty: 🟡 Medium Topics: Math, Number Theory Solution Problem Understanding The problem asks us to find the kth factor of a positive integer n , where the factors are ordered in ascending order. A factor of n is any integer i such that: $n \bmod i = 0$ For example, if n = 12 , the factors are: [1, 2, 3, 4, 6, 12] If k =...

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Codeforces Beta Round 17

Codeforces Beta Round 17 Type: Beta | Problems: 5 | Verified: 4/5 | Rating range: 1000-2900 | Time: 16m 51s Problem Name Rating Tags Solve Time Verified A Noldbach problem 1000 brute-force, math, number-theory 1m 46s ✓ B Hierarchy 1500 dfs-and-similar, dsu, greedy 1m 34s ✓ C Balance 2500 dp 6m 37s ✗ D Notepad 2400 number-theory 3m 4s ✓ E Palisection 2900 strings 3m 50s ✓

Practice › Codeforces Solutions › Codeforces Beta Round 61 (Div. 2) ›

CF 66D - Petya and His Friends

CF 66D - Petya and His Friends Rating: 1700 Tags: constructive algorithms, math, number theory Solve time: 1m 57s Verified: no Solution Problem Understanding We need to construct n distinct positive integers with two simultaneous properties. First, every pair of numbers must share a common divisor larger than 1 . In other words, for every pair (ai, aj) , their gcd cannot equal 1 . Second, the gcd of the...

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CF 7C - Line

CF 7C - Line Rating: 1800 Tags: math, number theory Solve time: 52s Verified: yes Solution Problem Understanding We are asked to find an integer point on a straight line described by the equation Ax + By + C = 0 . The inputs are three integers, A , B , and C , which define the slope and position of the line. Our task is to output integers x...

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Codeforces Beta Round 26 (Codeforces format)

Codeforces Beta Round 26 (Codeforces format) Type: Beta | Problems: 5 | Verified: 4/5 | Rating range: 900-2400 | Time: 7m 38s Problem Name Rating Tags Solve Time Verified A Almost Prime 900 number-theory 1m 18s ✓ B Regular Bracket Sequence 1400 greedy 1m 20s ✓ C Parquet 2000 combinatorics, constructive-algorithms, greedy 2m 9s ✗ D Tickets 2400 combinatorics, math, probabilities 1m 14s ✓ E Multithreading 2400 constructive-algorithms 1m 37s...

Practice › Codeforces Solutions › Codeforces Beta Round 17 ›

CF 17D - Notepad

CF 17D - Notepad Rating: 2400 Tags: number theory Solve time: 3m 4s Verified: yes Solution Problem Understanding Nick wants to list all numbers of a given length n in base b , where digits range from 0 to b-1 , but numbers cannot start with 0. Each page in his notepad holds exactly c numbers. We are asked to compute how many numbers appear on the last page he...

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LeetCode 1819 - Number of Different Subsequences GCDs

LeetCode Problem 1819 Difficulty: 🔴 Hard Topics: Array, Math, Counting, Number Theory Solution Problem Understanding The problem asks us to compute how many distinct values can appear as the GCD of some non-empty subsequence of the given array. A subsequence is formed by deleting zero or more elements while preserving order. Since the order does not matter for the GCD operation, the real challenge is determining which integers can be...

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CF 78C - Beaver Game

CF 78C - Beaver Game Rating: 2000 Tags: dp, games, number theory Solve time: 2m 6s Verified: yes Solution Problem Understanding We have n independent logs, each with length m . On a move, a player chooses one existing log and splits it into several equal pieces. If a log of length x is split into t equal parts, then t > 1 , t must divide x , and...

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CF 10C - Digital Root

CF 10C - Digital Root Rating: 2000 Tags: number theory Solve time: 1m 41s Verified: yes Solution Problem Understanding We are given an integer N . Consider all triples (A, B, C) where every value lies in the range [1, N] . Billy wants to verify whether A * B = C , but instead of multiplying directly, he compares digital roots: $$d(A \cdot B) = d(C)$$ Using the property...

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Codeforces Beta Round 91 (Div. 1 Only)

Codeforces Beta Round 91 (Div. 1 Only) Type: Div. 1 | Problems: 5 | Verified: 5/5 | Rating range: 1100-2500 | Time: 9m 56s Problem Name Rating Tags Solve Time Verified A Lucky Sum 1100 implementation 1m 36s ✓ B Lucky Transformation 1500 strings 2m 26s ✓ C Lucky Permutation 1900 brute-force, combinatorics, number-theory 1m 34s ✓ D Lucky Segments 2500 binary-search, implementation, two-pointers 1m 59s ✓ E Lucky Array...

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LeetCode 1799 - Maximize Score After N Operations

LeetCode Problem 1799 Difficulty: 🔴 Hard Topics: Array, Math, Dynamic Programming, Backtracking, Bit Manipulation, Number Theory, Bitmask Solution Problem Understanding The problem asks us to maximize the total score obtained by performing n operations on an array nums of size 2 * n . In each operation, you select two elements from the array, calculate the greatest common divisor (GCD) of the two elements, multiply it by the operation index...

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CF 64E - Prime Segment

CF 64E - Prime Segment Rating: 1800 Tags: *special, brute force, math, number theory Solve time: 1m 28s Verified: yes Solution Problem Understanding We are asked to find the closest prime numbers surrounding a given integer n . Concretely, we need two numbers, a and b , such that a ≤ n ≤ b and both a and b are prime. Among all prime intervals containing n , the one...

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Codeforces Beta Round 65 (Div. 2)

Codeforces Beta Round 65 (Div. 2) Type: Div. 2 | Problems: 5 | Verified: 4/5 | Rating range: 800-2200 | Time: 8m 23s Problem Name Rating Tags Solve Time Verified A Way Too Long Words 800 strings 1m 28s ✓ B Progress Bar 1300 implementation, math 1m 7s ✓ C Round Table Knights 1600 dp, math, number-theory 1m 40s ✓ D Solitaire 2200 brute-force, implementation 2m 18s ✓ E Nuclear...

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Codeforces Beta Round 88

Codeforces Beta Round 88 Type: Beta | Problems: 5 | Verified: 5/5 | Rating range: 1300-2900 | Time: 13m 35s Problem Name Rating Tags Solve Time Verified A Elevator 1300 implementation, math 2m 31s ✓ B Very Interesting Game 1800 brute-force, number-theory 2m 31s ✓ C Cycle 2000 dfs-and-similar, graphs 3m ✓ D Not Quick Transformation 2500 divide-and-conquer, math 2m 20s ✓ E Tree or not Tree 2900 data-structures, divide-and-conquer,...

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LeetCode 2001 - Number of Pairs of Interchangeable Rectangles

LeetCode Problem 2001 Difficulty: 🟡 Medium Topics: Array, Hash Table, Math, Counting, Number Theory Solution LeetCode 2001 - Number of Pairs of Interchangeable Rectangles Problem Understanding This problem asks us to count how many pairs of rectangles are interchangeable. Each rectangle is represented by two integers, width and height . Two rectangles are considered interchangeable if their width-to-height ratios are exactly the same. In mathematical terms, rectangles i and j...

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CF 93E - Lostborn

CF 93E - Lostborn Rating: 2600 Tags: dp, math, number theory Solve time: 1m 5s Verified: yes Solution Problem Understanding We are asked to calculate how many numbers from 1 to n are not divisible by any number in a given set of integers, called hit indicators. Each indicator is guaranteed to be coprime with the others, and there are k of them, each at most 1000. Conceptually, imagine Igor's...

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LeetCode 866 - Prime Palindrome

LeetCode Problem 866 Difficulty: 🟡 Medium Topics: Math, Number Theory Solution Problem Understanding This problem asks us to find the smallest number greater than or equal to n that satisfies two properties simultaneously: The number must be a palindrome. The number must be prime. A palindrome is a number that reads the same forward and backward. For example, 121 , 1331 , and 7 are palindromes. A prime number is...

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Codeforces Beta Round 72 (Div. 2 Only)

Codeforces Beta Round 72 (Div. 2 Only) Type: Div. 2 | Problems: 2 | Verified: 1/2 | Rating range: 900-1700 | Time: 4m 27s Problem Name Rating Tags Solve Time Verified A Toy Army 900 math, number-theory 1m 30s ✓ C Biathlon 1700 binary-search, implementation 2m 57s ✗

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LeetCode 1250 - Check If It Is a Good Array

LeetCode Problem 1250 Difficulty: 🔴 Hard Topics: Array, Math, Number Theory Solution Problem Understanding This problem asks whether it is possible to create the number 1 using the integers in the array, where each chosen number can be multiplied by any integer, including negative integers, before all results are summed together. More formally, given an array nums , we want to know whether there exist integers: $$x_1, x_2, x_3, \dots,...

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Codeforces Beta Round 51

Codeforces Beta Round 51 Type: Beta | Problems: 5 | Verified: 3/5 | Rating range: 1200-2500 | Time: 10m 12s Problem Name Rating Tags Solve Time Verified A Flea travel 1200 implementation, math 1m 24s ✓ B Smallest number 1600 brute-force 1m 47s ✓ C Pie or die 1900 games 1m 43s ✓ D Beautiful numbers 2500 dp, number-theory 3m 29s ✗ E Very simple problem 2500 geometry, two-pointers 1m...

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LeetCode 204 - Count Primes

LeetCode Problem 204 Difficulty: 🟡 Medium Topics: Array, Math, Enumeration, Number Theory Solution Problem Understanding The problem asks us to count how many prime numbers exist that are strictly smaller than a given integer n . A prime number is a positive integer greater than 1 that has exactly two divisors: 1 and itself. Examples of prime numbers include 2 , 3 , 5 , 7 , and 11 ....

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Appendix A.1 Sets and Functions

Sets A set is a collection of objects called elements. If $a$ is an element of a set $A$, we write $$ a\in A. $$ If $a$ is not an element of $A$, we write $$ a\notin A. $$ Sets are fundamental throughout mathematics because nearly every mathematical structure can be described in terms of collections and relations between collections. Examples include: $$ \mathbb{N}={1,2,3,\ldots}, $$ $$ \mathbb{Z}={\ldots,-2,-1,0,1,2,\ldots}, $$ and finite...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Future Directions in Number Theory

The Expanding Landscape Modern number theory continues to evolve rapidly. Classical themes such as primes, Diophantine equations, and reciprocity laws remain central, but the subject now interacts deeply with: computation, geometry, physics, probability, topology, cryptography, machine-assisted mathematics. Future developments will likely come not from a single direction, but from increasingly interconnected structures. Number theory has become both more abstract and more computational at the same time. Arithmetic Geometry Arithmetic geometry...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

The Langlands Program

A Grand Unifying Vision The Langlands program is one of the largest and most influential research programs in modern mathematics. Initiated by entity["people","Robert Langlands","Canadian mathematician"] in the late 1960s, it proposes deep correspondences connecting: number theory, representation theory, harmonic analysis, algebraic geometry, arithmetic geometry. At its core, the program predicts that arithmetic information carried by Galois groups corresponds to analytic information carried by automorphic forms and representations. The program is...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

The Birch and Swinnerton-Dyer Conjecture

Elliptic Curves and Rational Points An elliptic curve over $\mathbb{Q}$ may be written in Weierstrass form $$ E:y^2=x^3+ax+b, $$ where $$ 4a^3+27b^2\neq0. $$ The rational points $$ E(\mathbb{Q}) $$ form an abelian group. By the Mordell-Weil theorem, $$ E(\mathbb{Q}) \cong E(\mathbb{Q})_{\mathrm{tors}} \oplus \mathbb{Z}^r. $$ The finite subgroup $$ E(\mathbb{Q})_{\mathrm{tors}} $$ contains torsion points, while $$ r $$ is the rank. The rank measures the number of independent infinite-order rational points....

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

The Riemann Hypothesis

The Zeta Function The Riemann zeta function is one of the central objects in mathematics. For complex numbers $s$ with $$ \operatorname{Re}(s)>1, $$ it is defined by the convergent series $$ \zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s}. $$ It may also be written as the Euler product $$ \zeta(s) = \prod_p \left(1-\frac1{p^s}\right)^{-1}, $$ where the product runs over all primes. This identity connects the zeta function directly to the prime numbers. Analytic Continuation Although the...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Fermat's Last Theorem

The Statement Fermat's Last Theorem states that there are no positive integers $$ x,y,z $$ satisfying $$ x^n+y^n=z^n $$ when $$ n>2. $$ For $n=2$, the equation $$ x^2+y^2=z^2 $$ has many solutions, called Pythagorean triples. For example, $$ 3^2+4^2=5^2. $$ Fermat's Last Theorem says that this phenomenon stops completely for powers greater than $2$. Reduction to Prime Exponents It is enough to prove the theorem when $n$ is prime....

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Open Problems in Number Theory

The Nature of Open Problems Number theory contains some of the oldest and deepest unsolved problems in mathematics. Many are deceptively simple to state. Elementary questions about divisibility, primes, rational points, or polynomial equations often resist centuries of effort. Open problems serve several purposes: they organize research directions, they reveal structural gaps in current theory, they connect different branches of mathematics, they motivate new techniques and conjectures. Modern number theory...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Arithmetic Statistics

Statistics in Arithmetic Arithmetic statistics studies the distribution of arithmetic objects inside large families. Instead of analyzing a single integer, elliptic curve, number field, or modular form, one asks statistical questions: How common are certain properties? What does a typical object look like? Which invariants have limiting distributions? How often do exceptional behaviors occur? This subject combines number theory, probability, algebraic geometry, and random matrix theory. Arithmetic statistics is one...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Probabilistic Models for Primes

Primes as Rare Events Prime numbers are deterministic objects, but many aspects of their distribution resemble random behavior. The prime number theorem states that the number of primes up to $x$ satisfies $$ \pi(x)\sim \frac{x}{\log x}. $$ Equivalently, a large integer near $x$ behaves heuristically as if it were prime with probability $$ \frac1{\log x}. $$ This observation motivates probabilistic models for primes. Such models do not claim that primes...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Random Matrices and Zeta Zeros

Zeros as Spectral Data The Riemann zeta function is defined for $\operatorname{Re}(s)>1$ by $$ \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}. $$ It admits analytic continuation to most of the complex plane and satisfies a functional equation. Its nontrivial zeros lie in the critical strip $$ 0<\operatorname{Re}(s)<1. $$ The Riemann Hypothesis states that every nontrivial zero has the form $$ s=\frac12+i\gamma. $$ The numbers $\gamma$ may be viewed as spectral data. They behave, in many statistical...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Parity Problem

A Fundamental Limitation of Sieve Methods Sieve methods are extremely effective for estimating how many integers avoid small prime factors. They have produced major results about: almost primes, primes in arithmetic progressions, bounded gaps, additive prime problems. However, classical sieve methods encounter a deep structural obstacle known as the parity problem. Roughly speaking, sieves cannot reliably distinguish integers with: an even number of prime factors, an odd number of prime...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Chen's Theorem

Approximating the Twin Prime Conjecture The Twin Prime Conjecture states that infinitely many primes satisfy $$ p+2\text{ prime}. $$ Despite enormous effort, this remains open. One of the greatest achievements of sieve theory is Chen's theorem, proved by entity["people","Chen Jingrun","Chinese mathematician"] in 1973. Chen showed that infinitely many primes $p$ satisfy $$ p+2 $$ having at most two prime factors. Thus infinitely many numbers of the form $$ (p,p+2) $$...

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Probabilistic Algorithms

Randomness as a Computational Resource A probabilistic algorithm uses random choices during its execution. In number theory, this is often a practical advantage rather than a weakness. Many arithmetic problems have deterministic structure but become easier when one samples random residues, random group elements, random curves, or random auxiliary parameters. Randomness is useful because it can avoid worst-case patterns, find witnesses quickly, and replace exhaustive search with high-probability success. Monte...

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Bombieri-Vinogradov Theorem

Average Distribution of Primes The Prime Number Theorem for arithmetic progressions states that for $$ (a,q)=1, $$ the primes are asymptotically evenly distributed among residue classes modulo $q$: $$ \pi(x;q,a) \sim \frac{\operatorname{li}(x)}{\varphi(q)}. $$ However, proving strong error terms uniformly for large moduli $q$ is extremely difficult. The Generalized Riemann Hypothesis predicts essentially optimal uniform bounds. Remarkably, the Bombieri-Vinogradov theorem achieves GRH-level strength on average over moduli, without assuming GRH. This...

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Geometric Langlands Theory

From Arithmetic to Geometry The classical Langlands program relates: automorphic representations; Galois representations; $L$-functions. Its original setting is arithmetic: number fields, primes, and automorphic forms. Geometric Langlands theory transforms this picture into geometry. Instead of studying number fields, one studies algebraic curves over fields such as $$ \mathbb{C}. $$ Instead of automorphic functions, one studies sheaves and categories. Instead of Galois representations, one studies local systems and flat bundles. The...

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Hints for Selected Problems

Hints for Problem Set 1. Foundations 1. Sum of Two Even Integers Write the two integers as $$ a=2m,\qquad b=2n. $$ Then factor $2$ from $a+b$. 2. Product of Two Odd Integers Write $$ a=2m+1,\qquad b=2n+1. $$ Expand $ab$, then show it has the form $2k+1$. 3. If $n^2$ Is Even, Then $n$ Is Even Use proof by contrapositive. Assume $n$ is odd. Then write $$ n=2k+1. $$ Show that...

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Brun-Titchmarsh Theorem

Primes in Arithmetic Progressions One of the central problems of analytic number theory is understanding how primes distribute among residue classes. For integers $a$ and $q$ with $$ (a,q)=1, $$ define $$ \pi(x;q,a) = #{p\leq x : p\equiv a\pmod q}. $$ The Prime Number Theorem for arithmetic progressions states that $$ \pi(x;q,a) \sim \frac{\operatorname{li}(x)}{\varphi(q)}. $$ However, this asymptotic theorem becomes difficult to use when $q$ grows with $x$. The Brun-Titchmarsh...

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Probabilistic Primality

Deterministic and Probabilistic Testing A primality test determines whether an integer is prime. Some algorithms provide absolute certainty. These are deterministic primality tests. Other algorithms allow a small probability of error. These are probabilistic primality tests. In practice, probabilistic methods are often much faster and simpler. For cryptographic applications, an error probability smaller than hardware failure rates is usually acceptable. Modern computational number theory therefore relies heavily on probabilistic primality...

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Shimura Varieties

Higher-Dimensional Modular Curves Modular curves parameterize elliptic curves and connect modular forms with arithmetic geometry. Shimura varieties generalize this idea to higher dimensions. They are moduli spaces associated with more general algebraic groups and parameterize richer geometric structures such as: abelian varieties; Hodge structures; additional endomorphisms; level structures. Shimura varieties occupy a central position in the Langlands program because their geometry naturally carries both: automorphic information; Galois information. They are...

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Trace Formulas

Generalized Fourier Analysis Fourier analysis decomposes functions into harmonic frequencies. For functions on the circle, the exponential functions $$ e^{2\pi i n x} $$ form the spectral building blocks. Automorphic representation theory requires a vastly more sophisticated version of this idea. The trace formula is the fundamental tool for performing harmonic analysis on nonabelian arithmetic quotients. It relates: spectral information from automorphic representations; geometric information from conjugacy classes and orbital...

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Large Sieve

From Arithmetic Progressions to Harmonic Analysis Classical sieve methods estimate how many integers survive congruence restrictions. The large sieve approaches these problems from a different direction. Instead of counting integers directly, the method estimates the average size of exponential sums and character sums over many moduli simultaneously. The large sieve therefore connects: sieve theory, Fourier analysis, harmonic analysis on finite groups, distribution of arithmetic sequences. It became one of the...

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Frobenius Automorphisms

Arithmetic Symmetry of Primes One of the deepest ideas in algebraic number theory is that prime numbers possess hidden symmetry inside field extensions. Suppose $$ L/K $$ is a finite Galois extension of number fields with Galois group $$ G=\operatorname{Gal}(L/K). $$ Let $$ \mathfrak p $$ be a prime ideal of $$ \mathcal O_K, $$ and let $$ \mathfrak P $$ be a prime ideal of $$ \mathcal O_L $$...

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Decomposition and Inertia Groups

Galois Symmetry and Prime Ideals Let $$ L/K $$ be a finite Galois extension of number fields with Galois group $$ G=\operatorname{Gal}(L/K). $$ Suppose $$ \mathfrak p $$ is a prime ideal of $$ \mathcal O_K, $$ and let $$ \mathfrak P $$ be a prime ideal of $$ \mathcal O_L $$ lying above $\mathfrak p$, meaning $$ \mathfrak P\cap\mathcal O_K=\mathfrak p. $$ The Galois group acts on prime ideals:...

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Selberg Sieve

Refining Sieve Methods Brun's sieve introduced the idea of estimating sifted sets through truncated inclusion-exclusion. However, Brun's method often produced bounds that were technically difficult to optimize. entity["people","Atle Selberg","Norwegian mathematician"] developed a more flexible approach known as the Selberg sieve. Instead of truncating inclusion-exclusion directly, Selberg constructed carefully chosen weight functions that minimize error terms. The method became one of the central techniques of modern sieve theory. The Sifting Problem...

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Smooth Numbers

Definition A positive integer is called $y$-smooth if all of its prime factors are at most $y$. For example, $$ 72=2^3\cdot 3^2 $$ is $3$-smooth, because its only prime factors are $2$ and $3$. The integer $$ 210=2\cdot3\cdot5\cdot7 $$ is $7$-smooth, but not $5$-smooth. Smooth numbers are integers whose prime factors are small relative to their size. They form a sparse but highly structured subset of the positive integers. The...

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Automorphic $L$-Functions

Generalized Zeta Functions The Riemann zeta function $$ \zeta(s) = \sum_{n=1}^\infty \frac1{n^s} = \prod_p \frac1{1-p^{-s}} $$ encodes the arithmetic of prime numbers. Modern number theory studies vast generalizations of this function called automorphic $L$-functions. These functions arise from automorphic representations and modular forms. They unify many classical analytic objects, including: Dirichlet $L$-functions; Dedekind zeta functions; modular $L$-functions; Hasse-Weil $L$-functions. Automorphic $L$-functions lie at the center of the Langlands program. Euler...

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Brun Sieve

Sieve Methods Sieve methods are techniques for counting integers that remain after removing residue classes modulo primes. The basic idea is ancient. To find primes up to $x$, one removes multiples of small primes: $$ 2,3,5,7,\ldots $$ The remaining numbers have no small prime factors. This is the idea behind the sieve of Eratosthenes. Modern sieve theory turns this procedure into a quantitative analytic method. It estimates the size of...

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Ramification of Primes

Prime Factorization in Extensions In the ordinary integers, every nonzero integer factors uniquely into prime numbers. In algebraic number theory, one studies how ordinary primes behave inside the ring of integers of a number field. Let $$ K/\mathbb{Q} $$ be a number field extension, and let $$ \mathcal O_K $$ be its ring of integers. A rational prime $$ p $$ generates the ideal $$ (p)=p\mathcal O_K. $$ Unlike ordinary...

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Random Integers

Arithmetic as a Probabilistic Object Number theory often studies exact statements about individual integers. For example, one may ask whether a given integer is prime, squarefree, smooth, or representable as a sum of squares. Probabilistic number theory asks a different kind of question: what does a typical integer look like? Instead of studying one integer at a time, we study large intervals such as $$ {1,2,\ldots,N} $$ and examine what...

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Functoriality

Transfers Between Symmetries The Langlands program predicts that many different arithmetic objects are connected by systematic transfers. These transfers are called functoriality. At a basic level, functoriality says that a relationship between symmetry groups should produce a relationship between automorphic representations. This principle turns scattered identities in number theory into instances of one general mechanism. Class field theory, base change, modularity, and many lifting theorems can all be viewed as...

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Schnirelmann Density

Measuring Additive Size In additive number theory, ordinary asymptotic density is often too weak to control additive behavior. For example, a set may have density zero yet still represent all sufficiently large integers after repeated addition. Prime numbers are an important example. To study additive growth more effectively, entity["people","Lev Schnirelmann","Russian mathematician"] introduced a stronger density notion adapted specifically to additive problems. This density became one of the foundational tools of...

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Local Fields

Localizing Arithmetic Classical number theory studies arithmetic globally over fields such as $$ \mathbb{Q} $$ or number fields. A local field studies arithmetic near a single prime or near infinity. Instead of considering all divisibility simultaneously, one focuses on one valuation at a time. This local viewpoint simplifies many problems and reveals deep structural phenomena invisible globally. The central examples are: the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, the...

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Zero-Knowledge Proofs

Proving Without Revealing A zero-knowledge proof allows one party to convince another that a statement is true without revealing why it is true. This idea appears paradoxical at first. Traditional proofs reveal logical structure and intermediate reasoning. Zero-knowledge proofs instead reveal only the validity of the statement. The subject combines: computational complexity, number theory, algebra, probability, cryptography. Zero-knowledge protocols are now central in privacy-preserving systems, authentication, blockchains, and modern cryptographic...

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Galois Representations

Arithmetic Symmetry as Linear Algebra Galois groups encode the symmetries of algebraic equations and field extensions. Unfortunately, absolute Galois groups are usually enormous and extremely complicated. For example, $$ \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) $$ is a profinite group of immense complexity. To study such groups, one translates their abstract symmetry into linear algebra. This leads to Galois representations. A Galois representation converts arithmetic symmetry into matrices acting on vector spaces. Modern number theory...

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Additive Bases

Representing Integers by Sums A central question in additive number theory asks whether every integer can be represented as a sum of elements from a fixed set. Let $$ A\subseteq \mathbb N. $$ We say that $A$ is an additive basis if sufficiently large integers can be written as sums of finitely many elements of $A$. The subject studies how additive structure emerges from repeated summation of sets. Basis of...

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$p$-Adic Numbers

A New Arithmetic Geometry The real numbers arise by completing the rational numbers using the ordinary absolute value. The $p$-adic numbers arise by completing the rational numbers using the $p$-adic absolute value. This produces a radically different geometry. In the real numbers, powers of large integers grow without bound: $$ 2^n\to\infty. $$ In the $2$-adic world, $$ 2^n\to0. $$ The geometry is governed not by magnitude but by divisibility. The...

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Post-Quantum Cryptography

The Quantum Threat Modern public-key cryptography relies heavily on two computational assumptions: factoring large integers is hard, discrete logarithms are hard. RSA depends on factoring. Diffie-Hellman and elliptic curve cryptography depend on discrete logarithms. Quantum computation changes this landscape dramatically. entity["people","Peter Shor","American mathematician"] discovered algorithms showing that sufficiently powerful quantum computers could solve both problems in polynomial time. A large fault-tolerant quantum computer would therefore break much of current public-key...

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The Langlands Program

A Grand Unification The Langlands program is one of the most ambitious and influential theories in modern mathematics. Proposed by entity["people","Robert Langlands","Canadian mathematician"] in the late 1960s, it predicts deep correspondences between: automorphic representations; Galois representations; $L$-functions; harmonic analysis; arithmetic geometry. At its core lies a unifying philosophy: Arithmetic symmetry and analytic symmetry are manifestations of the same underlying structure. The program generalizes class field theory from abelian extensions to...

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Exponential Sums

Oscillation in Number Theory Exponential sums are among the central tools of analytic number theory. They measure cancellation in oscillatory arithmetic expressions and provide quantitative control over additive structure. A typical exponential sum has the form $$ \sum_{n} e^{2\pi i f(n)}, $$ where $f(n)$ is a real-valued arithmetic function. Because complex exponentials oscillate around the unit circle, large cancellation often occurs. Understanding this cancellation is fundamental in problems involving primes,...

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Valuations and Absolute Values

Measuring Arithmetic Size In ordinary analysis, the absolute value $$ |x| $$ measures the size of a real number. In number theory, one studies more general notions of size that reflect arithmetic divisibility rather than geometric distance. These generalized size functions are called valuations or absolute values. They allow arithmetic to be studied analytically and lead naturally to $p$-adic numbers, local fields, and modern arithmetic geometry. Ordinary Absolute Value The...

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Dedekind Domains

Arithmetic Beyond Unique Factorization Ordinary integers satisfy several remarkable properties simultaneously: every ideal is principal, every nonzero element factors uniquely into primes, every nonzero ideal factors uniquely into prime ideals. When algebraic number theory developed, mathematicians discovered that many rings of algebraic integers lose some of these properties. Unique factorization of elements may fail, and ideals need not be principal. Nevertheless, rings of integers still possess enough structure to support...

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Suggested Projects and Explorations

Project 1. Prime Number Experiments Goals Study empirical properties of prime numbers through computation. Possible Tasks Implement the sieve of Eratosthenes. Plot the prime counting function $$ \pi(x). $$ Compare $$ \pi(x) $$ with $$ \frac{x}{\log x}. $$ Investigate prime gaps. Search for twin primes. Compute distributions of primes modulo small integers. Extensions Study the Chebyshev functions. Experiment with probabilistic prime models. Compare actual prime counts with logarithmic integral approximations....

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Lattice Cryptography

Cryptography from Lattice Problems Lattice cryptography is a family of cryptographic systems based on the presumed hardness of computational problems on high-dimensional lattices. A lattice is a discrete set of points in Euclidean space generated by integer combinations of basis vectors: $$ L=\mathbb{Z}b_1+\cdots+\mathbb{Z}b_n. $$ The security of lattice cryptography comes from the difficulty of problems such as finding short vectors, finding close lattice points, or recovering hidden linear structure in...

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Circle Method

Additive Representation Problems Many problems in additive number theory ask whether an integer can be represented in the form $$ n=a_1+a_2+\cdots+a_k, $$ where the variables satisfy arithmetic restrictions. Examples include: Goldbach-type problems, Waring's problem, representations by quadratic forms, additive problems involving primes. The Hardy-Littlewood circle method is one of the most powerful techniques for studying such questions. The method transforms arithmetic counting problems into problems of harmonic analysis on the...

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Automorphic Representations

From Functions to Representations Classical modular form theory begins with analytic functions satisfying symmetry conditions. Modern theory shifts perspective. Instead of studying individual functions directly, one studies the representations generated by these functions under group actions. This transition from functions to representations is one of the defining conceptual advances of twentieth-century mathematics. Automorphic representations provide the natural language for: automorphic forms; Hecke operators; $L$-functions; the Langlands program. They unify harmonic...

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Principal Ideals

Ideals Generated by One Element Let $R$ be a commutative ring. An ideal $I\subseteq R$ is called principal if there exists an element $\alpha\in R$ such that $$ I=(\alpha). $$ This notation means $$ (\alpha)={\alpha r:r\in R}. $$ Thus a principal ideal consists of all multiples of a single element. In the ordinary integers, $$ (n)=n\mathbb{Z} $$ is the set of all multiples of $n$. For example, $$ (6)={\dots,-12,-6,0,6,12,\dots}. $$...

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Waring's Problem

Sums of Powers Waring's problem asks whether every sufficiently large positive integer can be written as a sum of a bounded number of fixed powers. For example, one may ask whether every large integer can be written as a sum of squares, $$ n=x_1^2+x_2^2+\cdots+x_k^2, $$ or as a sum of cubes, $$ n=x_1^3+x_2^3+\cdots+x_k^3. $$ Here the exponent is fixed, while the number of summands is allowed to depend on that...

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Pairing-Based Cryptography

Bilinear Pairings Pairing-based cryptography uses special maps defined on elliptic curve groups. A pairing is a function $$ e:G_1\times G_2\to G_T, $$ where $G_1$, $G_2$, and $G_T$ are finite groups, usually of the same large prime order. The key property is bilinearity. For all integers $a,b$ and group elements $P\in G_1$, $Q\in G_2$, $$ e([a]P,[b]Q)=e(P,Q)^{ab}. $$ This identity makes pairings useful in cryptography. It allows multiplicative relations among hidden exponents...

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Automorphic Forms

Beyond Modular Forms Modular forms are functions on the upper half-plane satisfying symmetry conditions under the modular group $$ SL_2(\mathbb{Z}). $$ They encode deep arithmetic information and connect complex analysis with number theory. Automorphic forms generalize this idea. Instead of studying functions invariant under subgroups of $$ SL_2(\mathbb{Z}), $$ one studies functions on much more general groups and symmetric spaces. This broader theory unifies: modular forms; representation theory; harmonic analysis;...

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Discriminants

Measuring Arithmetic Complexity The discriminant is one of the most important invariants of a number field. It measures how the arithmetic of the field differs from ordinary rational arithmetic. Roughly speaking, the discriminant detects: linear dependence among embeddings, ramification of primes, geometric density of the ring of integers, arithmetic complexity of the field. It appears throughout algebraic number theory, arithmetic geometry, and analytic number theory. Discriminant of a Polynomial Let...

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Goldbach Problems

Additive Questions About Primes Goldbach-type problems ask whether integers can be represented as sums of primes. They are among the oldest and most famous problems in additive number theory. The guiding question is simple: $$ n=p_1+p_2+\cdots+p_k, $$ where the $p_i$ are prime numbers. Such questions connect two different aspects of primes. Multiplicatively, primes are the building blocks of integers. Additively, they appear to behave like a dense enough set to...

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Elliptic Curve Cryptography

From Finite Fields to Elliptic Curves Elliptic curve cryptography is a public-key cryptographic framework based on the arithmetic of elliptic curves over finite fields. Classical Diffie-Hellman uses exponentiation in a finite cyclic group. Elliptic curve cryptography replaces this with scalar multiplication on an elliptic curve group. Instead of computing $$ g^n, $$ one computes $$ [n]P. $$ Here $P$ is a point on an elliptic curve, and $[n]P$ means adding...

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The Modularity Theorem

From Elliptic Curves to Modular Forms For centuries, elliptic curves and modular forms were studied as separate objects. Elliptic curves arose from algebraic equations and Diophantine problems. Modular forms arose from complex analysis and modular symmetry. The modularity theorem revealed that these two theories are fundamentally the same. Every elliptic curve over $\mathbb{Q}$ corresponds to a modular form. This theorem is one of the deepest achievements of twentieth-century mathematics and...

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Units and Dirichlet Unit Theorem

Multiplicative Invertibility Let $K$ be a number field and let $$ \mathcal O_K $$ be its ring of integers. An element $$ u\in\mathcal O_K $$ is called a unit if it has a multiplicative inverse inside $\mathcal O_K$. That is, there exists $$ v\in\mathcal O_K $$ such that $$ uv=1. $$ The set of all units forms a multiplicative group denoted $$ \mathcal O_K^\times. $$ Units generalize the familiar invertible...

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Class Groups

Measuring Failure of Unique Factorization In ordinary integers, every ideal is generated by a single element: $$ I=(n). $$ This property is closely related to unique factorization. In rings of integers of number fields, not every ideal is principal. The obstruction to principality is measured by the class group. The class group is one of the central invariants of algebraic number theory. It measures how far a ring of integers...

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Diffie-Hellman Key Exchange

The Key Exchange Problem Secure communication requires two parties to share secret information. In classical symmetric cryptography, both parties must already possess the same secret key before communication begins. This raises a practical problem: how can two people agree on a secret key over an insecure channel? The Diffie-Hellman key exchange solves this problem. It allows two parties to create a shared secret even when an attacker can observe all...

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Sumsets

Addition of Sets Additive number theory studies arithmetic structure through addition of integers and subsets of integers. Given two subsets $$ A,B\subseteq \mathbb Z, $$ their sumset is defined by $$ A+B = {a+b : a\in A,\ b\in B}. $$ Thus the sumset contains every integer obtainable as a sum of one element from $A$ and one element from $B$. Sumsets measure additive structure and form one of the central...

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Elliptic Curves and Modularity

Elliptic Curves as Arithmetic Objects An elliptic curve is simultaneously: an algebraic curve; a complex torus; an abelian group; an arithmetic object. These curves lie at the center of modern number theory. They appear in: Diophantine equations; cryptography; modular forms; Galois representations; the Langlands program. One of the deepest discoveries of twentieth-century mathematics is that elliptic curves over $\mathbb{Q}$ are governed by modular forms. This relationship is called modularity. Algebraic...

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Problem Sets

Problem Set 1. Foundations Prove that the sum of two even integers is even. Prove that the product of two odd integers is odd. Prove that if $n^2$ is even, then $n$ is even. Prove by induction that $$ 1+2+\cdots+n=\frac{n(n+1)}{2}. $$ Prove by induction that $$ 1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}. $$ Prove that every integer $n\ge2$ has a prime divisor. Prove that there are infinitely many primes. Show that if $a\mid b$ and...

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Generalized Riemann Hypothesis

Extending the Riemann Hypothesis The classical Riemann Hypothesis concerns the zeros of the Riemann zeta function $$ \zeta(s). $$ However, modern number theory studies many other $L$-functions, especially Dirichlet $L$-functions $$ L(s,\chi). $$ These functions possess analytic continuation, Euler products, and functional equations analogous to those of the zeta function. The Generalized Riemann Hypothesis (GRH) extends Riemann’s conjecture to this broader family. Statement of GRH Let $$ \chi $$ be...

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Ideals and Prime Ideals

Motivation from Factorization Failure In ordinary integers, every number factors uniquely into primes. In many rings of algebraic integers, this property fails. For example, in $$ \mathbb{Z}[\sqrt{-5}], $$ one has $$ 6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5}), $$ with genuinely different factorizations. This failure makes divisibility difficult to control if one works only with elements. The key insight of modern algebraic number theory is that factorization becomes unique again when one replaces elements by ideals....

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Unique Factorization Failure

Unique Factorization in the Integers One of the central properties of the ordinary integers is unique factorization. Every nonzero integer can be written as a product of prime numbers, and this factorization is unique up to order and signs. For example, $$ 60=2^2\cdot3\cdot5. $$ No essentially different prime factorization exists. This theorem, called the fundamental theorem of arithmetic, underlies much of classical number theory. When arithmetic is extended to algebraic...

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Norm and Trace

Two Fundamental Invariants Let $K$ be a number field of degree $$ [K:\mathbb{Q}]=n. $$ Each element $\alpha\in K$ has two fundamental rational invariants: its norm and its trace. These generalize familiar operations from quadratic fields. The norm behaves like a product of conjugates. The trace behaves like a sum of conjugates. If the embeddings of $K$ into $\mathbb{C}$ are $$ \sigma_1,\sigma_2,\ldots,\sigma_n, $$ then the norm and trace are defined by...

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RSA Cryptosystem

Public-Key Cryptography Classical cryptography uses a shared secret key. Both sender and receiver must know the same secret information in advance. Public-key cryptography introduced a radically different idea: encryption uses a public key, decryption uses a private key. Anyone may encrypt messages using the public key, but only the holder of the private key can decrypt them. The RSA cryptosystem, introduced by entity["people","Ron Rivest","American cryptographer"], entity["people","Adi Shamir","Israeli cryptographer"], and entity["people","Leonard...

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Modular Curves

Geometry of Modular Symmetry The modular group acts on the upper half-plane by fractional linear transformations: $$ z\mapsto \frac{az+b}{cz+d}. $$ This action identifies points related by modular symmetry. Taking the quotient produces geometric spaces called modular curves. Although constructed analytically, modular curves are fundamentally arithmetic objects. They simultaneously encode: complex analysis; algebraic geometry; elliptic curves; modular forms; Galois actions. Modern arithmetic geometry treats modular curves as one of the most...

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Nonvanishing Results

Importance of Nonvanishing A central theme in analytic number theory is determining when an $L$-function is nonzero at a particular point. Nonvanishing results are important because zeros of $L$-functions encode arithmetic information. If an $L$-function vanishes unexpectedly, strong cancellation occurs inside the associated arithmetic data. The classical example is Dirichlet’s theorem on primes in arithmetic progressions, whose proof depends critically on the fact that $$ L(1,\chi)\neq0 $$ for every nonprincipal...

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Symbolic and Numeric Computation

Computation in Modern Number Theory Modern number theory relies heavily on computation. Two broad computational paradigms dominate the subject: symbolic computation, numeric computation. Symbolic computation manipulates exact mathematical expressions such as integers, polynomials, algebraic numbers, and formal power series. Numeric computation approximates mathematical quantities using floating-point arithmetic and analytic methods. Both approaches are essential. Symbolic methods preserve exactness, while numerical methods allow efficient approximation of quantities too complicated for exact...

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Primes in Arithmetic Progressions

Arithmetic Progressions An arithmetic progression is a sequence of the form $$ a,\ a+q,\ a+2q,\ a+3q,\ \ldots, $$ where $a$ and $q$ are fixed integers with $$ q>0. $$ A fundamental question in number theory asks: Are there infinitely many primes in a given arithmetic progression? For example: primes congruent to $1\pmod4$, primes congruent to $3\pmod4$, primes congruent to $1\pmod3$. Some progressions obviously contain only finitely many primes. For instance,...

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Hecke Operators

Symmetries of Modular Forms Modular forms already possess symmetry under the modular group. Yet a deeper arithmetic structure emerges through another family of operators: the Hecke operators. These operators act linearly on spaces of modular forms and reveal hidden multiplicative structure in Fourier coefficients. The theory of Hecke operators transformed modular forms from analytic curiosities into arithmetic objects connected with: prime numbers; $L$-functions; elliptic curves; Galois representations; automorphic forms. Modern...

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Ring of Integers

Integers Inside a Number Field In ordinary arithmetic, the integers $$ \mathbb{Z} $$ form the fundamental arithmetic structure inside the rational numbers $\mathbb{Q}$. For a number field $K$, the analogous object is its ring of integers, denoted $$ \mathcal O_K. $$ The ring of integers consists of all algebraic integers contained in $K$: $$ \mathcal O_K = {\alpha\in K:\alpha \text{ is an algebraic integer}}. $$ This ring generalizes ordinary integers...

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Dirichlet $L$-Functions

Generalizing the Zeta Function The Riemann zeta function $$ \zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s} $$ studies primes globally, without distinguishing residue classes modulo $q$. To study primes in arithmetic progressions, entity["people","Peter Gustav Lejeune Dirichlet","German mathematician"] introduced a family of functions built from Dirichlet characters. These are the Dirichlet $L$-functions. They are among the most important objects in analytic number theory and provide the analytic foundation for Dirichlet’s theorem on primes in arithmetic...

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Algorithms for Elliptic Curves

Elliptic Curves as Computational Objects Elliptic curves occupy a central position in modern number theory, arithmetic geometry, and cryptography. An elliptic curve over a field $K$ is typically given by a Weierstrass equation $$ y^2=x^3+ax+b, $$ where $$ 4a^3+27b^2\neq0. $$ The nonvanishing condition ensures smoothness. Elliptic curves are simultaneously: algebraic curves, abelian groups, arithmetic objects, cryptographic platforms. Because of this rich structure, many efficient computational algorithms have been developed for...

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Cusp Forms

Vanishing at the Cusps Modular forms satisfy strong symmetry conditions under the modular group. Among them, cusp forms form the deepest and most arithmetic subclass. A cusp form is a modular form that vanishes at every cusp of the modular curve. While Eisenstein series reflect the “continuous” part of modular form theory, cusp forms behave more discretely and mysteriously. Their Fourier coefficients encode subtle arithmetic information and are closely connected...

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Algorithms for Modular Forms

Modular Forms as Computable Objects Modular forms are highly structured analytic functions with deep arithmetic properties. Although their definitions involve complex analysis and group actions, modular forms are also computable objects. Modern computational number theory studies algorithms for: computing Fourier expansions, evaluating Hecke operators, determining eigenforms, constructing modular curves, computing associated $L$-functions, constructing Galois representations. These algorithms are essential in arithmetic geometry, cryptography, and modern algebraic number theory. Fourier Expansions...

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Eisenstein Series

The Simplest Modular Forms Among all modular forms, Eisenstein series are the most explicit and computationally accessible. They arise naturally by summing simple rational functions over lattice points in the plane. Despite their elementary definition, they encode deep arithmetic information. Their Fourier coefficients involve divisor sums, their products generate large parts of the algebra of modular forms, and their analytic properties connect modular form theory with zeta functions and $L$-functions....

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Orthogonality Relations

Characters as Harmonic Functions Dirichlet characters behave analogously to exponential functions in Fourier analysis. Just as complex exponentials separate frequencies, characters separate residue classes modulo $q$. The key mechanism behind this separation is orthogonality. Orthogonality relations allow sums over characters to isolate specific congruence conditions. They are among the most important computational tools in analytic number theory. Characters Modulo $q$ Fix a positive integer $q$. Let $$ \chi \pmod q...

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Number Fields

Fields Generated by Algebraic Numbers A number field is a finite extension of the rational numbers. Concretely, it is a field $K$ satisfying $$ \mathbb{Q}\subseteq K\subseteq \mathbb{C} $$ and such that $K$ has finite dimension as a vector space over $\mathbb{Q}$. The simplest examples are obtained by adjoining an algebraic number $\alpha$ to $\mathbb{Q}$. The resulting field is denoted $$ \mathbb{Q}(\alpha). $$ It is the smallest field containing both $\mathbb{Q}$...

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Modular Forms

Analytic Functions with Arithmetic Symmetry Modular forms are among the central objects of modern number theory. At first sight, they are simply holomorphic functions on the upper half-plane satisfying special transformation rules under the modular group. Yet these functions encode profound arithmetic information. Their Fourier coefficients appear in: partition formulas; counting problems; elliptic curves; $L$-functions; Galois representations. The proof of entity["people","Andrew Wiles","British mathematician"] of entity["historical_event","Fermat's Last Theorem","1994 proof by Andrew...

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Lattice Reduction

Lattices A lattice is a discrete additive subgroup of Euclidean space. More concretely, let $$ b_1,\ldots,b_n\in\mathbb{R}^m $$ be linearly independent vectors. The lattice generated by these vectors is $$ L=\mathbb{Z}b_1+\cdots+\mathbb{Z}b_n. $$ Thus a lattice consists of all integer linear combinations $$ a_1b_1+\cdots+a_nb_n, \qquad a_i\in\mathbb{Z}. $$ The vectors $$ b_1,\ldots,b_n $$ form a basis of the lattice. A lattice may have many different bases. Some bases are nearly orthogonal and contain...

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Chronology of Number Theory

Ancient Mathematics Period Development c. 1800 BCE Babylonian arithmetic tables and quadratic problems c. 1650 BCE Egyptian arithmetic in the Rhind Papyrus c. 500 BCE Early Greek studies of ratios and integers c. 300 BCE entity["people","Euclid","ancient Greek mathematician"] writes Elements, including Euclidean algorithm and infinitude of primes c. 250 BCE Study of perfect numbers and geometric arithmetic Classical and Late Ancient Era Period Development c. 250 CE entity["people","Diophantus","ancient Greek mathematician"]...

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Dirichlet Characters

Motivation The Riemann zeta function studies prime numbers globally, without distinguishing congruence classes. However, many arithmetic questions concern primes satisfying conditions such as $$ p\equiv a\pmod q. $$ For example: Are there infinitely many primes congruent to $1\pmod4$? How are primes distributed modulo $q$? Do different residue classes contain approximately the same number of primes? To study such questions analytically, entity["people","Peter Gustav Lejeune Dirichlet","German mathematician"] introduced Dirichlet characters. These functions...

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Modular Functions

Functions Invariant Under Modular Symmetry The modular group acts on the upper half-plane by fractional linear transformations: $$ z\mapsto \frac{az+b}{cz+d}. $$ Modular forms transform predictably under this action. A modular function is even more symmetric: it remains invariant. These functions play a central role in complex analysis, algebraic geometry, and number theory. They provide coordinates on modular curves and encode deep arithmetic information. The most important example is the $j$-invariant,...

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Minimal Polynomials

Algebraic Dependence Over $\mathbb{Q}$ An algebraic number is a complex number that satisfies some nonzero polynomial equation with rational coefficients. Thus $\alpha\in\mathbb{C}$ is algebraic if there exists a nonzero polynomial $$ f(x)\in\mathbb{Q}[x] $$ such that $$ f(\alpha)=0. $$ Among all such polynomials, one is distinguished: the monic polynomial of least degree. This polynomial is called the minimal polynomial of $\alpha$ over $\mathbb{Q}$. The minimal polynomial records the simplest exact algebraic...

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Index of Definitions

Foundations Definition Location Natural Number 1.1 Integer 1.2 Absolute Value 1.5 Mathematical Induction 1.6 Strong Induction 1.7 Recursive Definition 1.8 Divisibility Definition Location Divisibility 2.1 Prime Number 2.2 Composite Number 2.3 Greatest Common Divisor 2.5 Least Common Multiple 2.6 Coprime Integers 2.10 Prime Factorization Definition Location Prime Factorization 3.1 Canonical Prime Decomposition 3.2 Prime Counting Function 12.1 Prime Gap 12.7 Congruences Definition Location Congruence Modulo $n$ 4.1 Residue Class 4.2...

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Connections with Prime Distribution

Prime Numbers and Analytic Structure The Riemann zeta function was introduced through the series $$ \zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s}, $$ but its true importance comes from its connection with prime numbers. Through Euler’s product formula, $$ \zeta(s) = \prod_p \left(1-\frac1{p^s}\right)^{-1}, $$ the arithmetic structure of the primes becomes encoded inside an analytic function. Analytic properties of $\zeta(s)$ therefore translate into information about prime distribution. This principle forms the foundation of analytic...

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Integer Factorization

The Factorization Problem Integer factorization asks for the prime decomposition of a positive integer. Given $$ n>1, $$ we seek primes $$ p_1,\ldots,p_r $$ and positive exponents $$ e_1,\ldots,e_r $$ such that $$ n=p_1^{e_1}\cdots p_r^{e_r}. $$ The fundamental theorem of arithmetic guarantees that this decomposition exists and is unique up to the order of the factors. Computationally, however, finding the factors may be difficult. This distinction between existence and efficient...

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Euler Products

From Sums to Products Euler products are one of the central ideas of analytic number theory. They express infinite sums over integers as infinite products over primes. The basic principle comes from unique prime factorization. Since every positive integer decomposes uniquely into primes, multiplicative information about integers can often be reorganized prime by prime. The simplest and most important example is the Euler product for the Riemann zeta function: $$...

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Dirichlet Series

Infinite Series Attached to Arithmetic Functions An arithmetic function $f(n)$ can be encoded into an infinite series of the form $$ \sum_{n=1}^{\infty}\frac{f(n)}{n^s}, $$ where $s$ is usually a real or complex variable. Such a series is called a Dirichlet series. Dirichlet series are fundamental because they transform arithmetic information into analytic information. Multiplicative structure becomes product structure, and divisor sums become ordinary multiplication of series. General Form A Dirichlet series...

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Modular Groups

Fractional Linear Transformations Modular forms begin with the action of certain matrix groups on the complex upper half-plane. Let $$ \mathbb{H} = {z\in\mathbb{C}:\operatorname{Im}(z)>0}. $$ The upper half-plane is a natural domain for complex analysis and hyperbolic geometry. A matrix $$ \gamma= \begin{pmatrix} a & b\ c & d \end{pmatrix} $$ with real entries and determinant $ad-bc\neq0$ acts on $z\in\mathbb{H}$ by $$ \gamma z = \frac{az+b}{cz+d}. $$ This is called a...

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Algebraic Integers

Beyond the Ordinary Integers The ordinary integers $$ \mathbb{Z}={\dots,-2,-1,0,1,2,\dots} $$ form the basic arithmetic system of number theory. However, many Diophantine equations naturally lead to larger number systems. For example, the equation $$ x^2+1=0 $$ has no solution in $\mathbb{Z}$ or $\mathbb{R}$, but it has solutions in the complex numbers: $$ x=\pm i. $$ Similarly, Pell equations lead naturally to expressions involving $$ \sqrt D. $$ To study arithmetic systematically...

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Index of Theorems

Elementary Number Theory Theorem Location Division Algorithm 2.4 Euclidean Algorithm 2.7 Extended Euclidean Algorithm 2.8 Bézout Identity 2.9 Euclid’s Lemma 3.1 Fundamental Theorem of Arithmetic 3.1 Infinitude of Primes 3.4 Chinese Remainder Theorem 4.7 Euler’s Theorem 5.3 Möbius Inversion Formula 5.7 Diophantine Equations Theorem Location Solvability of Linear Diophantine Equations 6.1 Classification of Primitive Pythagorean Triples 6.2 Pell Equation Structure Theorem 6.3 Fermat Two-Squares Theorem 6.4 Rational Point Criterion for...

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Primality Testing

The Problem of Recognizing Primes A prime number is an integer greater than $1$ whose only positive divisors are $$ 1 \quad\text{and}\quad p. $$ Prime numbers are the fundamental building blocks of arithmetic because every integer factors uniquely into primes. One of the oldest computational problems in number theory is therefore: Given an integer $n$, determine whether $n$ is prime. This is the primality testing problem. For small integers, direct...

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Global Class Field Theory

The Classification Problem One of the central goals of algebraic number theory is to classify field extensions of a number field $$ K. $$ Galois theory reduces this problem to understanding Galois groups. Among all extensions, the abelian extensions are the first major class that admits a complete description. Global class field theory provides this description. It explains all finite abelian extensions of a number field in terms of arithmetic...

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Explicit Formulae

From Zeros to Prime Numbers One of the deepest ideas in analytic number theory is that the zeros of the zeta function determine the distribution of prime numbers. This relationship is expressed through explicit formulae, which connect arithmetic counting functions with sums over zeta zeros. These formulas make precise the principle that fluctuations in prime distribution arise from oscillations produced by complex zeros. Chebyshev Functions The most natural setting for...

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Diophantine Approximation

Approximating Real Numbers by Rational Numbers Diophantine approximation studies how closely real numbers can be approximated by rational numbers. Given a real number $\alpha$, one seeks rational fractions $$ \frac pq $$ such that $$ \left| \alpha-\frac pq \right| $$ is very small. The subject lies between number theory, analysis, and geometry. It investigates how arithmetic structure constrains approximation quality. Rational Numbers and Density The rational numbers are dense in...

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Average Orders of Arithmetic Functions

Irregular Arithmetic Functions Arithmetic functions often fluctuate strongly from one integer to the next. For example, $$ \tau(p)=2 $$ for every prime $p$, while $$ \tau(360)=24. $$ Similarly, $$ \varphi(p)=p-1 $$ for primes, but $$ \varphi(2^k)=2^{k-1}. $$ Because of this irregular behavior, exact formulas for individual integers are often difficult to analyze globally. Instead of studying precise values, number theory frequently studies average behavior. Average Order Let $f(n)$ be an...

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Möbius Inversion

Recovering a Function from Divisor Sums Many arithmetic functions are defined through sums over divisors. For example, $$ \sigma(n)=\sum_{d\mid n} d $$ and $$ \tau(n)=\sum_{d\mid n}1. $$ A natural question arises: if a function $F(n)$ is defined by summing another function $f(n)$ over the divisors of $n$, can one recover $f$ from $F$? Möbius inversion gives the answer. Statement of Möbius Inversion Suppose $f$ and $F$ are arithmetic functions satisfying...

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Pell Equations via Continued Fractions

Pell Equations Revisited Recall that a Pell equation has the form $$ x^2-Dy^2=1, $$ where $D$ is a positive nonsquare integer. $$ x^2-Dy^2=1 $$ Earlier we saw that such equations possess infinitely many integer solutions. Continued fractions provide the systematic method for finding them. The key fact is that the continued fraction expansion of $$ \sqrt D $$ is periodic. This periodicity produces rational approximations $$ \frac pq $$ for...

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Riemann Hypothesis

Statement of the Conjecture The Riemann zeta function has nontrivial zeros inside the critical strip $$ 0<\operatorname{Re}(s)<1. $$ The Riemann Hypothesis asserts that every nontrivial zero lies on the critical line $$ \operatorname{Re}(s)=\frac12. $$ Thus if $$ \zeta(\rho)=0 $$ and $\rho$ is nontrivial, then $$ \rho=\frac12+it $$ for some real number $t$. This conjecture was proposed by entity["people","Bernhard Riemann","German mathematician"] in his 1859 memoir on prime numbers. It is widely...

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Fast Integer Arithmetic

Arithmetic Complexity Modern computational number theory depends fundamentally on efficient arithmetic with large integers. Elementary school algorithms for addition and multiplication are sufficient for small numbers, but modern applications involve integers with hundreds, thousands, or millions of digits. Examples include: cryptography, primality testing, integer factorization, computational algebra, symbolic computation. The efficiency of arithmetic algorithms therefore becomes a central concern. The cost of arithmetic operations is measured asymptotically in terms of...

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Local Class Field Theory

From Global Arithmetic to Local Arithmetic Global class field theory studies finite abelian extensions of number fields such as $$ \mathbb{Q} \quad\text{or}\quad \mathbb{Q}(\sqrt d). $$ Its central result is that abelian Galois groups are controlled by arithmetic objects built from ideals and ideles. Local class field theory develops the corresponding theory for local fields: $$ \mathbb{Q}_p, \qquad \mathbb{R}, \qquad \mathbb{C}, $$ and finite extensions of $\mathbb{Q}_p$. The local theory is...

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Dirichlet Convolution

Combining Arithmetic Functions Arithmetic functions can be added and multiplied pointwise, but number theory has another product that is better adapted to divisibility. Let $f$ and $g$ be arithmetic functions. Their Dirichlet convolution is the arithmetic function $f*g$ defined by $$ (f*g)(n)=\sum_{d\mid n} f(d)g\left(\frac nd\right). $$ The sum runs over all positive divisors $d$ of $n$. This operation combines the values of $f$ and $g$ over the divisor structure of...

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Convergents

Successive Rational Approximations The convergents of a continued fraction are the rational numbers obtained by truncating the expansion at finite stages. If $$ \alpha=[a_0;a_1,a_2,a_3,\dots], $$ then the $n$-th convergent is $$ \frac{p_n}{q_n} = [a_0;a_1,\dots,a_n]. $$ Convergents are among the best possible rational approximations to $\alpha$. They encode the arithmetic structure of the continued fraction expansion. First Examples Consider $$ \sqrt2=[1;\overline2]. $$ Its convergents are $$ 1, \frac32, \frac75, \frac{17}{12}, \frac{41}{29},...

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Zeros of the Zeta Function

The Equation $\zeta(s)=0$ The zeros of the Riemann zeta function are the complex numbers $s$ satisfying $$ \zeta(s)=0. $$ These zeros are among the most important objects in mathematics because they govern the fine structure of prime distribution. The study of zeta zeros lies at the center of analytic number theory, connecting complex analysis, harmonic analysis, probability, spectral theory, and arithmetic. Trivial Zeros The functional equation $$ \zeta(s) = 2^s\pi^{s-1}...

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Notation Index

Number Systems Symbol Meaning $\mathbb{N}$ natural numbers $\mathbb{Z}$ integers $\mathbb{Q}$ rational numbers $\mathbb{R}$ real numbers $\mathbb{C}$ complex numbers $\mathbb{F}_p$ finite field with $p$ elements $\mathbb{Q}_p$ field of $p$-adic numbers $\mathbb{Z}_p$ ring of $p$-adic integers Sets and Logic Symbol Meaning $x\in A$ $x$ belongs to $A$ $x\notin A$ $x$ does not belong to $A$ $A\subseteq B$ $A$ is a subset of $B$ $A\cup B$ union $A\cap B$ intersection $A\setminus B$ set...

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Trace Formula

Spectral Decomposition and Arithmetic One of the central ideas of modern analysis is that functions may be decomposed spectrally into elementary pieces. Fourier analysis decomposes periodic functions into exponential waves: $$ e^{2\pi i nx}. $$ Representation theory generalizes this philosophy to noncommutative groups. For arithmetic groups and adelic groups, the corresponding decomposition is governed by automorphic representations. The trace formula is a tool that compares: spectral information from representations, geometric...

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Hilbert Class Fields

Failure of Unique Factorization One of the central discoveries of algebraic number theory is that unique factorization may fail in rings of algebraic integers. For example, in the ring $$ \mathbb{Z}[\sqrt{-5}], $$ the number $6$ factors in two essentially different ways: $$ 6=2\cdot3 $$ and $$ 6=(1+\sqrt{-5})(1-\sqrt{-5}). $$ None of these factors are associates, and all are irreducible. Thus unique factorization breaks down. Dedekind resolved this difficulty by replacing factorization...

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Functional Equation

Symmetry of the Zeta Function The defining series of the zeta function, $$ \zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s}, $$ converges only in the half-plane $$ \operatorname{Re}(s)>1. $$ Analytic continuation extends the function to the entire complex plane except for a pole at $s=1$. However, the continuation alone does not reveal the deep internal symmetry of the zeta function. This symmetry is expressed by the functional equation, discovered by entity["people","Bernhard Riemann","German mathematician"]. The...

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Rational Approximations

Approximating Irrational Numbers Many important numbers are irrational: $$ \sqrt2,\qquad \pi,\qquad e. $$ Since irrational numbers cannot be written exactly as fractions, one seeks rational approximations $$ \frac pq $$ that are close to the target number. The central question is: How well can irrational numbers be approximated by rational numbers? Continued fractions provide the most systematic answer to this problem. Measuring Approximation Error Suppose $\alpha$ is a real number...

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Completely Multiplicative Functions

Multiplicative and Completely Multiplicative Functions An arithmetic function is a function defined on the positive integers. Such a function $$ f:\mathbb{N}\to\mathbb{C} $$ is called multiplicative if $$ f(ab)=f(a)f(b) $$ whenever $$ \gcd(a,b)=1. $$ It is called completely multiplicative if the same identity holds for all positive integers $a$ and $b$, without assuming coprimality: $$ f(ab)=f(a)f(b). $$ Thus complete multiplicativity is stronger than multiplicativity. Basic Examples The constant function $$ f(n)=1...

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Liouville Function

Prime Factors with Multiplicity The Liouville function is an arithmetic function denoted by $$ \lambda(n). $$ It is defined using the total number of prime factors of $n$, counted with multiplicity. If $$ n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r} $$ is the canonical prime factorization of $n$, define $$ \Omega(n)=\alpha_1+\alpha_2+\cdots+\alpha_r. $$ Then the Liouville function is $$ \lambda(n)=(-1)^{\Omega(n)}. $$ Thus $\lambda(n)$ is $1$ when $n$ has an even number of prime factors counted with...

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Analytic Continuation

Beyond the Region of Convergence The defining series of the Riemann zeta function is $$ \zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s}, $$ which converges absolutely only when $$ \operatorname{Re}(s)>1. $$ At first sight, this seems to restrict the zeta function to the right half-plane. However, many important arithmetic phenomena occur outside this region. The central idea of analytic continuation is that a function initially defined on one domain may extend uniquely to a...

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Reciprocity Maps

Reciprocity in Arithmetic One of the oldest themes in number theory is reciprocity: the phenomenon that solvability conditions for one prime are controlled by arithmetic involving another prime. The classical example is quadratic reciprocity, discovered by entity["people","Carl Friedrich Gauss","German mathematician"]. It describes when a prime number $p$ is a quadratic residue modulo another prime $q$. For odd primes $p$ and $q$, $$ \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{(p-1)(q-1)}4}. $$ At first sight,...

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Infinite Continued Fractions

Irrational Numbers and Infinite Expansions Finite continued fractions correspond exactly to rational numbers. When the Euclidean algorithm never terminates, the continued fraction becomes infinite. An infinite continued fraction has the form $$ [a_0;a_1,a_2,a_3,\ldots], $$ where $$ a_0\in\mathbb{Z}, \qquad a_i\in\mathbb{Z}_{>0} \text{ for } i\ge1. $$ Such expressions represent irrational numbers. For example, $$ \sqrt2=[1;2,2,2,\ldots]. $$ $$ \sqrt{2}=[1;2,2,2,\ldots] $$ Infinite continued fractions provide one of the most natural arithmetic descriptions of irrational...

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Functoriality

The Central Principle of the Langlands Program Functoriality is the unifying mechanism of the Langlands program. It predicts systematic relationships between automorphic representations attached to different algebraic groups. The principle may be summarized informally as follows: homomorphisms between symmetry groups should produce transfers between automorphic representations. This idea organizes vast families of correspondences and explains many identities between $L$-functions. Functoriality is considered one of the deepest conjectural structures in modern...

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Euler Totient Function

Counting Coprime Residues Euler's totient function is an arithmetic function denoted by $$ \varphi(n). $$ For a positive integer $n$, it counts the integers $a$ satisfying $$ 1\le a\le n $$ and $$ \gcd(a,n)=1. $$ Thus $$ \varphi(n)=#{a:1\le a\le n,\ \gcd(a,n)=1}. $$ For example, the integers from $1$ to $10$ that are coprime to $10$ are $$ 1,3,7,9. $$ Therefore $$ \varphi(10)=4. $$ The totient function measures how many residue...

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Euler Product Formula

Arithmetic Structure of the Zeta Function The defining series of the Riemann zeta function is $$ \zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s}, \qquad \operatorname{Re}(s)>1. $$ At first sight, this appears to be merely an analytic object: an infinite series over the positive integers. Its true arithmetic significance emerges through the Euler product formula, which expresses the same function as a product over primes. This identity is one of the foundational results of analytic...

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Abelian Extensions

Field Extensions and Commutative Symmetry A central goal of algebraic number theory is to understand field extensions of a given base field, especially extensions of the rational numbers $$ \mathbb{Q}. $$ Galois theory associates to every finite Galois extension $$ L/K $$ a group of symmetries: $$ \mathrm{Gal}(L/K). $$ The structure of this group reflects the arithmetic complexity of the extension. Among all extensions, the simplest and most important are...

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Finite Continued Fractions

From Division to Expansion A finite continued fraction is an expression of the form $$ a_0+\frac{1}{a_1+\frac{1}{a_2+\cdots+\frac{1}{a_n}}}, $$ where $$ a_0\in\mathbb{Z} $$ and $$ a_1,a_2,\ldots,a_n\in\mathbb{Z}_{>0}. $$ It is usually written compactly as $$ [a_0;a_1,a_2,\ldots,a_n]. $$ Finite continued fractions are another way to write rational numbers. They arise directly from the Euclidean algorithm. First Examples Consider $$ [2;3]. $$ This means $$ 2+\frac13=\frac73. $$ Now consider $$ [1;2,2]. $$ Then $$ [1;2,2]...

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Glossary

A Abelian Group A group $G$ is abelian if $$ ab=ba $$ for all $a,b\in G$. Examples include $(\mathbb{Z},+)$ and $(\mathbb{Q}^{\times},\cdot)$. Absolute Value For a real number $x$, $$ |x|= \begin{cases} x & x\ge0, \ -x & x<0. \end{cases} $$ For a complex number $z=a+bi$, $$ |z|=\sqrt{a^2+b^2}. $$ Adeles The adele ring combines all completions of a global field into a single topological ring. Adeles unify archimedean and nonarchimedean arithmetic....

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Definition of the Zeta Function

Infinite Series and Prime Numbers One of the central objects of analytic number theory is the Riemann zeta function. It connects infinite series, prime numbers, complex analysis, and arithmetic structure into a single analytic framework. The zeta function begins with the series $$ \sum_{n=1}^{\infty}\frac1{n^s}, $$ where $s$ is a complex variable. This function first appeared in the work of entity["people","Leonhard Euler","Swiss mathematician"] during the study of infinite series. Later, entity["people","Bernhard...

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Euclidean Algorithm Revisited

Division and Remainders The Euclidean algorithm is one of the oldest and most important algorithms in mathematics. It computes the greatest common divisor of two integers using repeated division. Given integers $a$ and $b$ with $$ a>b>0, $$ the division algorithm states that there exist unique integers $q$ and $r$ such that $$ a=bq+r, \qquad 0\le r<b. $$ $$ a=bq+r $$ The Euclidean algorithm repeatedly applies this decomposition. For example,...

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Adeles and Ideles

Simultaneous Arithmetic at All Places The rational numbers may be studied through their completions: $$ \mathbb{R}, \qquad \mathbb{Q}_2, \qquad \mathbb{Q}_3, \qquad \mathbb{Q}_5, \ldots $$ Each completion captures arithmetic near one place: $\mathbb{R}$ describes archimedean behavior; $\mathbb{Q}_p$ describes behavior near the prime $p$. Modern number theory seeks a framework in which all completions are considered simultaneously. Adeles and ideles provide this framework. They unify local and global arithmetic into a single...

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Möbius Function

Definition The Möbius function is an arithmetic function denoted by $$ \mu(n). $$ It is defined from the prime factorization of $n$. For $n=1$, $$ \mu(1)=1. $$ For $n>1$, write $$ n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}. $$ Then $$ \mu(n)= \begin{cases} 0, & \text{if some } \alpha_i\ge2,\ (-1)^r, & \text{if every } \alpha_i=1. \end{cases} $$ Thus $\mu(n)$ detects whether $n$ is squarefree, and if it is squarefree, records the parity of the number...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Langlands Program

A Vast Web of Correspondences The Langlands program is a broad collection of conjectures connecting number theory, representation theory, harmonic analysis, and algebraic geometry. Its central idea is that two very different kinds of objects should secretly encode the same arithmetic information. On one side are Galois representations, which come from field extensions, algebraic varieties, and étale cohomology. On the other side are automorphic representations, which come from harmonic analysis...

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Twin Prime Heuristics

Twin Primes A pair of primes $$ (p,p+2) $$ is called a twin prime pair. Examples include $$ (3,5),\quad (5,7),\quad (11,13),\quad (17,19). $$ The Twin Prime Conjecture states that infinitely many such pairs exist. Despite enormous numerical evidence and major advances in sieve theory, the conjecture remains unproved. The goal of twin prime heuristics is to estimate how frequently such pairs should occur. Naive Probability Model The Prime Number Theorem...

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Computational Aspects

Quadratic Residues as Computational Objects Quadratic residue theory is not only a theoretical subject. It also plays a major role in computational number theory, cryptography, primality testing, and algorithm design. Many arithmetic algorithms reduce to questions of the form: $$ x^2\equiv a\pmod n. $$ $$ x^2\equiv a\pmod n $$ Typical computational tasks include: deciding whether $a$ is a quadratic residue modulo $n$, finding square roots modulo primes, computing Legendre or...

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Local-Global Principles

From Local Information to Global Solutions A central problem in number theory is determining whether an equation possesses rational or integral solutions. For example: $$ x^2+y^2=z^2, $$ $$ x^3+y^3=z^3, $$ or more generally, $$ f(x_1,\ldots,x_n)=0. $$ Such equations are called Diophantine equations. Directly finding rational solutions is often extremely difficult. A powerful strategy is to study the equation locally, meaning over completions of the rational numbers: $$ \mathbb{R}, \qquad \mathbb{Q}_2,...

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Appendix J. Historical Notes and Bibliography

J.1 Why History Matters Number theory is one of the oldest parts of mathematics, but modern number theory is not a single ancient subject carried forward unchanged. It is a layered discipline. Elementary divisibility, Diophantine equations, algebraic number theory, analytic number theory, modular forms, arithmetic geometry, and computational number theory arose at different times from different problems. Historical study helps explain why the subject has its present shape. The concepts...

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Divisor Functions

Counting and Summing Divisors Divisor functions measure the positive divisors of an integer. They are among the first examples of arithmetic functions, because their values depend directly on the prime factorization of the input. For a positive integer $n$, the most basic divisor functions are $$ \tau(n)=#{d\in\mathbb{N}:d\mid n} $$ and $$ \sigma(n)=\sum_{d\mid n} d. $$ The function $\tau(n)$ counts the positive divisors of $n$. The function $\sigma(n)$ sums them. For...

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Applications to Computation

Modular Arithmetic as Computation Modular arithmetic is not only a theoretical language for divisibility. It is also one of the main tools of computation with integers. A congruence $$ a\equiv b\pmod n $$ allows a large integer $a$ to be replaced by a smaller representative $b$ without losing information modulo $n$. This is useful because many computations depend only on remainders. For example, instead of carrying the large number $$...

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Prime Gaps

Consecutive Prime Differences Let $$ p_n $$ denote the $n$-th prime number. The difference $$ g_n=p_{n+1}-p_n $$ is called the $n$-th prime gap. For example, $$ 2,3,5,7,11,13 $$ produce the gaps $$ 1,2,2,4,2. $$ Prime gaps measure how irregularly primes are distributed among the integers. Average Size of Prime Gaps The Prime Number Theorem implies that primes near $x$ have average density approximately $$ \frac1{\log x}. $$ Thus the average...

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Adelic Methods

Local and Global Arithmetic Number theory studies arithmetic simultaneously at two levels: global arithmetic over number fields such as $\mathbb{Q}$, local arithmetic over completions such as $\mathbb{R}$ and $\mathbb{Q}_p$. Historically, these settings were studied separately. Adelic methods unify them into a single framework. The central idea is that arithmetic objects should be analyzed through all completions of a field at once. This philosophy lies at the heart of modern automorphic...

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Higher Reciprocity Laws

Beyond Quadratic Reciprocity Quadratic reciprocity describes when one prime is a square modulo another prime. A natural question is whether similar laws exist for higher powers. For example, one may ask: when is $a$ a cube modulo $p$? when is $a$ a fourth power modulo $p$? when does $$ x^n\equiv a\pmod p $$ have a solution? These questions lead to higher reciprocity laws, which generalize quadratic reciprocity to higher powers....

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Hensel’s Lemma

Lifting Solutions from Modular Arithmetic One of the central ideas of number theory is that congruences modulo powers of a prime often approximate genuine arithmetic solutions. Suppose a polynomial equation has a solution modulo $p$. Can this solution be refined to a solution modulo $p^2$, then modulo $p^3$, and eventually to a solution in the field of $p$-adic numbers? Hensel’s lemma answers this question. It is one of the foundational...

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Fast Modular Exponentiation

The Problem Modular arithmetic often requires computing powers such as $$ a^k \pmod n. $$ If $k$ is large, direct multiplication is inefficient. For example, computing $$ 7^{1000}\pmod{13} $$ by multiplying $7$ one thousand times is unnecessary. Fast modular exponentiation computes powers by repeatedly squaring and reducing modulo $n$. The method is efficient because it uses the binary expansion of the exponent. Repeated Squaring The basic idea is that powers...

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Appendix I. Computational Tools

I.1 Computation in Modern Number Theory Computation has become an essential part of number theory. Classical arithmetic relied mainly on symbolic reasoning and hand calculations. Modern arithmetic combines rigorous proof with large-scale computation, algorithmic experimentation, symbolic algebra, and numerical analysis. Computational methods are now central in: primality testing, integer factorization, cryptography, modular forms, elliptic curves, lattice reduction, arithmetic geometry, experimental mathematics. Large computations frequently guide conjectures before proofs are discovered....

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Short Intervals

Prime Distribution on Small Scales The Prime Number Theorem describes the average distribution of primes up to a large number $x$: $$ \pi(x)\sim \frac{x}{\log x}. $$ From this, one expects that an interval of length $h$ near $x$ should contain roughly $$ \frac{h}{\log x} $$ primes. Thus the quantity $$ \pi(x+h)-\pi(x) $$ should approximately equal $$ \frac{h}{\log x}. $$ The study of such questions is called the theory of primes...

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Automorphic Representations

From Automorphic Forms to Representations Classically, number theory studied special analytic functions such as modular forms. These functions satisfy strong symmetry conditions under actions of arithmetic groups. Modern theory reformulates these objects representation-theoretically. Instead of studying individual functions directly, one studies the spaces generated by their symmetries. The resulting objects are called automorphic representations. This shift from functions to representations is one of the major conceptual advances of twentieth-century mathematics....

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Completion of Fields

Incompleteness of the Rational Numbers The rational numbers form a field rich enough for arithmetic, yet insufficient for many limiting processes. Consider the sequence of decimal approximations to $\sqrt2$: $$ 1,\quad 1.4,\quad 1.41,\quad 1.414,\quad\ldots $$ The terms become arbitrarily close to one another, but the sequence does not converge inside $\mathbb{Q}$, because $$ \sqrt2\notin\mathbb{Q}. $$ Thus the rational numbers contain “holes.” The process of filling these holes leads to the...

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Chinese Remainder Theorem

Independent Congruence Conditions The Chinese remainder theorem describes when several congruence conditions can be combined into one congruence. Its cleanest form occurs when the moduli are pairwise coprime. Suppose $$ n_1,n_2,\ldots,n_r $$ are positive integers satisfying $$ \gcd(n_i,n_j)=1 $$ whenever $$ i\ne j. $$ Then the congruences $$ x\equiv a_1\pmod{n_1}, $$ $$ x\equiv a_2\pmod{n_2}, $$ $$ \cdots $$ $$ x\equiv a_r\pmod{n_r} $$ have a unique solution modulo $$ N=n_1n_2\cdots n_r....

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Error Terms

Measuring the Accuracy of Prime Approximations The Prime Number Theorem states that $$ \pi(x)\sim \operatorname{li}(x), $$ or equivalently, $$ \pi(x)\sim \frac{x}{\log x}. $$ These asymptotic formulas describe the main growth of the prime counting function. However, they do not explain how large the difference between the approximation and the true value may be. To study the precision of prime estimates, one introduces error terms. Basic Error Functions The two standard...

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Representation Theory Background

Symmetry and Linear Actions Representation theory studies abstract algebraic objects by expressing them as linear transformations of vector spaces. The guiding principle is simple: complicated algebraic structures become more understandable when realized concretely as matrices. If $G$ is a group and $V$ is a vector space over a field $k$, a representation of $G$ on $V$ is a homomorphism $$ \rho:G\to\operatorname{GL}(V), $$ where $$ \operatorname{GL}(V) $$ is the group of...

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$p$-Adic Numbers

Beyond the Real Numbers The real numbers arise by completing the rational numbers with respect to the ordinary absolute value. This completion produces a field suited to Euclidean geometry and classical analysis. Number theory reveals another completion process based on divisibility by a prime number $p$. The resulting fields are the $p$-adic numbers. While real analysis measures magnitude geometrically, $p$-adic analysis measures arithmetic proximity. Two numbers are close $p$-adically when...

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Gauss Sums

Characters and Exponential Sums Gauss sums arise from combining multiplicative and additive structures modulo a prime. They form one of the fundamental tools of analytic and algebraic number theory. Let $p$ be an odd prime and let $$ \chi(a)=\left(\frac{a}{p}\right) $$ be the quadratic character modulo $p$. Thus $$ \chi(a)= \begin{cases} 1 & \text{if } a \text{ is a quadratic residue modulo } p,\ -1 & \text{if } a \text{ is...

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Appendix H. Category Theory Basics

H.1 Motivation Category theory studies mathematical structures through objects and maps between them. Instead of looking only at what objects are made of, it studies how they relate to other objects. This viewpoint appears naturally in modern number theory. Groups, rings, fields, modules, schemes, varieties, sheaves, and representations all form categories. Many deep constructions are best understood as maps preserving structure. Category theory provides a common language for algebra, topology,...

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Weil Conjectures

Counting Points over Finite Fields One of the central problems in arithmetic geometry is understanding the number of solutions of polynomial equations over finite fields. Let $$ X $$ be an algebraic variety over the finite field $$ \mathbb{F}_q. $$ For each integer $n\geq1$, one may count the points of $X$ defined over the extension field $$ \mathbb{F}_{q^n}. $$ These numbers, $$ #X(\mathbb{F}_{q^n}), $$ contain deep arithmetic information. Examples quickly...

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Absolute Values

Measuring Size in Arithmetic The ordinary absolute value on the real numbers measures magnitude: $$ |x|= \begin{cases} x,&x\ge0,\ -x,&x<0. \end{cases} $$ This function allows us to define distance, $$ d(x,y)=|x-y|, $$ and therefore notions such as convergence, continuity, and completeness. In number theory, however, there are many different ways to measure the size of numbers. The ordinary absolute value is only one possibility. The theory of absolute values provides a...

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Quadratic Reciprocity

The Central Problem The theory of quadratic residues asks a fundamental question: Given distinct odd primes $p$ and $q$, when does the congruence $$ x^2\equiv q\pmod p $$ have a solution? Equivalently, when is $$ \left(\frac{q}{p}\right)=1? $$ At first glance, the residue behavior of $q$ modulo $p$ and the residue behavior of $p$ modulo $q$ appear unrelated. The law of quadratic reciprocity reveals a remarkable symmetry between them. This theorem...

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Systems of Congruences

Several Congruences at Once A system of congruences asks for an integer satisfying several congruence conditions simultaneously. For example, $$ x\equiv2\pmod3 $$ and $$ x\equiv3\pmod5 $$ form a system. A solution is an integer $x$ that satisfies both congruences. Testing small values, we find $$ 8\equiv2\pmod3 $$ and $$ 8\equiv3\pmod5. $$ Thus $$ x=8 $$ is one solution. Since congruence conditions repeat periodically, there are infinitely many integer solutions, but...

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Modular Inverses

Division in Modular Arithmetic In ordinary arithmetic, division by a nonzero number means multiplication by its reciprocal. Modular arithmetic is more delicate. A residue class may or may not have a multiplicative inverse. Let $a$ and $n$ be integers with $n\ge2$. A modular inverse of $a$ modulo $n$ is an integer $b$ satisfying $$ ab\equiv1\pmod n. $$ When such a $b$ exists, we write $$ b\equiv a^{-1}\pmod n. $$ This...

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Linear Congruences

Definition A linear congruence is a congruence of the form $$ ax\equiv b\pmod n, $$ where $a,b\in\mathbb{Z}$, $n\in\mathbb{N}$, and $x$ is the unknown integer. This congruence means that $$ n\mid(ax-b). $$ Equivalently, there exists an integer $y$ such that $$ ax-b=ny. $$ Rearranging gives the linear Diophantine equation $$ ax-ny=b. $$ Thus solving a linear congruence is closely related to solving a linear equation in two integer variables. First Examples...

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Arithmetic Modulo $n$

Modular Arithmetic Arithmetic modulo $n$ is arithmetic performed on residue classes modulo $n$. Instead of distinguishing all integers separately, we identify integers that have the same remainder after division by $n$. Thus, modulo $n$, every integer is represented by one of $$ 0,1,2,\ldots,n-1. $$ For example, modulo $5$, $$ 7\equiv2\pmod5, \qquad 13\equiv3\pmod5. $$ So in arithmetic modulo $5$, the integer $7$ behaves like $2$, and the integer $13$ behaves like...

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Residue Classes

From Congruence to Classes Congruence modulo $n$ groups integers according to their remainders after division by $n$. If two integers have the same remainder, they are congruent modulo $n$. For example, $$ 17\equiv2\pmod5 $$ because $$ 17-2=15 $$ and $$ 5\mid15. $$ The integers congruent to $2$ modulo $5$ form the set $$ {\ldots,-8,-3,2,7,12,17,\ldots}. $$ This set is called a residue class modulo $5$. Definition Let $n\in\mathbb{N}$. For an integer...

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Congruence Relations

Motivation Ordinary equality compares integers exactly. In many arithmetic problems, however, only the remainder after division matters. For example, the integers $$ 17 \quad\text{and}\quad 5 $$ leave the same remainder when divided by $12$: $$ 17=12\cdot1+5, \qquad 5=12\cdot0+5. $$ From the viewpoint of remainders modulo $12$, these integers behave identically. This observation leads to the notion of congruence. Definition of Congruence Let $a,b\in\mathbb{Z}$, and let $n\in\mathbb{N}$. We say that $a$...

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Distribution Heuristics of Primes

The Problem of Distribution The infinitude of primes guarantees that primes continue indefinitely, but it says nothing about how frequently primes occur. At first glance, primes appear irregular: $$ 2,3,5,7,11,13,17,19,23,29,\ldots $$ Some gaps are small: $$ 11,13. $$ Others are larger: $$ 23,29. $$ As integers grow, the pattern becomes increasingly difficult to predict. The study of how primes are distributed among the integers is one of the central themes...

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Euler's Proof

From Euclid to Euler Euclid proved that there are infinitely many primes by contradiction. Euler discovered a very different proof based on infinite series and products. His argument connects prime numbers with the harmonic series $$ \sum_{n=1}^{\infty}\frac1n. $$ This series diverges: $$ 1+\frac12+\frac13+\frac14+\cdots=\infty. $$ Euler observed that the harmonic series can be decomposed multiplicatively using prime factorization. This insight became one of the foundational ideas of analytic number theory. The...

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Euclid's Proof

The Classical Argument Euclid's proof of the infinitude of primes is one of the earliest examples of a general argument in number theory. It does not depend on computation, experimentation, or knowledge of large primes. It proves that no finite list of primes can ever be complete. Suppose we are given finitely many primes: $$ p_1,p_2,\ldots,p_k. $$ We do not yet assume that these are all primes. We merely ask...

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Infinitude of Primes

The Basic Question Prime numbers are the building blocks of the positive integers. Once unique prime factorization is known, a natural question arises: are there only finitely many primes, or do primes continue forever? The answer is that there are infinitely many primes. This result is one of the oldest theorems in number theory. Its proof is short, but the idea is powerful: any finite list of primes can be...

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Arithmetic Functions from Factorization

Functions on Positive Integers An arithmetic function is a function whose domain is the positive integers. It assigns a value to each integer $$ n\in\mathbb{N}. $$ Typical examples include the number of positive divisors of $n$, the sum of positive divisors of $n$, and the number of integers less than or equal to $n$ that are coprime to $n$. Prime factorization makes many arithmetic functions computable. Once we know $$...

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Canonical Prime Decomposition

Standard Form of an Integer Unique prime factorization says that every integer $n>1$ can be written as a product of primes. The canonical prime decomposition is the ordered and exponentiated version of this factorization. For every integer $n>1$, there exist distinct primes $$ p_1<p_2<\cdots<p_r $$ and positive integers $$ \alpha_1,\alpha_2,\ldots,\alpha_r $$ such that $$ n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}. $$ This expression is canonical because the primes are written in increasing order and...

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Appendix G. Linear Algebra Review

G.1 Vector Spaces A vector space over a field $F$ is a set $V$ equipped with addition and scalar multiplication satisfying the usual algebraic rules. The scalars belong to $F$, while the vectors belong to $V$. Examples: $$ \mathbb{R}^n,\qquad \mathbb{C}^n,\qquad \mathbb{Q}^n. $$ Polynomial spaces such as $$ F[x] $$ are also vector spaces over $F$. In number theory, vector spaces appear in: modular forms, representation theory, cohomology, field extensions, lattice...

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Logarithmic Integral

Definition The logarithmic integral is the function $$ \operatorname{li}(x) = \int_2^x \frac{dt}{\log t}. $$ It plays a central role in prime number theory because it gives a remarkably accurate approximation to the prime counting function $$ \pi(x). $$ The Prime Number Theorem may be written in the form $$ \pi(x)\sim \operatorname{li}(x). $$ Although the simpler approximation $$ \frac{x}{\log x} $$ captures the correct order of growth, the logarithmic integral incorporates...

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Étale Cohomology

The Need for a New Cohomology Theory Classical topology studies geometric spaces using invariants such as homology and cohomology. Over the complex numbers, algebraic varieties can often be viewed as topological spaces, and topological methods become available. However, arithmetic geometry studies varieties over fields such as $$ \mathbb{Q},\qquad \mathbb{F}_p,\qquad \mathbb{Q}_p, $$ where ordinary topology is insufficient or meaningless. For example, a variety over a finite field has only finitely many...

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Ramification

Primes in Field Extensions One of the central ideas of algebraic number theory is that prime numbers may behave differently after passing to a larger field. In the ordinary integers $\mathbb{Z}$, every nonzero integer factors uniquely into prime numbers. In a number field $K$, the role of prime numbers is played by prime ideals in the ring of integers $\mathcal{O}_K$. When extending fields, $$ \mathbb{Q}\subseteq K, $$ a prime number...

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Prime Number Theorem

Statement of the Theorem The Prime Number Theorem describes the asymptotic distribution of prime numbers. It states that $$ \pi(x)\sim \frac{x}{\log x} $$ as $x\to\infty$. Equivalently, $$ \lim_{x\to\infty}\frac{\pi(x)\log x}{x}=1. $$ Thus the density of primes near a large number $x$ is approximately $$ \frac1{\log x}. $$ The theorem gives a precise quantitative form to the observation that primes become less frequent as numbers grow larger. Logarithmic Integral Form A more...

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Appendix F. Measure and Integration

F.1 Motivation Measure theory extends the ideas of length, area, volume, and integration to more general settings. In number theory, measure appears in probability, harmonic analysis, ergodic theory, automorphic forms, adelic groups, and equidistribution. Classical integration over intervals is not sufficient for modern analytic methods. One needs integration on abstract spaces such as compact groups, local fields, and quotient spaces. Measure theory provides a rigorous framework for averaging arithmetic objects....

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Euler Criterion

Detecting Quadratic Residues Euler criterion gives an efficient way to decide whether an integer is a square modulo an odd prime. Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Then $a$ is either a quadratic residue modulo $p$ or a quadratic nonresidue modulo $p$. The criterion states that this distinction is detected by the power $$ a^{(p-1)/2}. $$ More precisely, $$ a^{(p-1)/2}\equiv...

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Arithmetic Surfaces

From Curves to Surfaces Arithmetic geometry often studies families of algebraic curves varying over arithmetic bases. The most important base is $$ \operatorname{Spec}(\mathbb{Z}), $$ whose points correspond to prime numbers together with the generic point associated to $\mathbb{Q}$. When a curve is extended from a field to the ring of integers, the resulting object typically becomes two-dimensional. Such objects are called arithmetic surfaces. An arithmetic surface is, roughly, a two-dimensional...

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Cyclotomic Fields

Roots of Unity One of the most important classes of number fields arises from the solutions of the equation $$ x^n=1. $$ The solutions are called the $n$-th roots of unity. In the complex numbers, these roots are $$ 1,\zeta_n,\zeta_n^2,\ldots,\zeta_n^{n-1}, $$ where $$ \zeta_n=e^{2\pi i/n}. $$ Geometrically, these numbers lie equally spaced around the unit circle in the complex plane. For example, the fourth roots of unity are $$ 1,\quad...

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Chebyshev Bounds

Motivation The prime counting function $$ \pi(x) $$ measures the number of primes not exceeding $x$. Numerical evidence suggests that $$ \pi(x)\approx \frac{x}{\log x}, $$ but proving such a statement is difficult. Before the Prime Number Theorem was established, Pafnuty Chebyshev obtained strong upper and lower bounds showing that the order of growth of $\pi(x)$ is indeed approximately $x/\log x$. These estimates were among the first major successes of analytic...

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Jacobi Symbol

Motivation The Legendre symbol $$ \left(\frac{a}{p}\right) $$ is defined only when the denominator is an odd prime. In many problems, however, one must work modulo an odd composite integer. The Jacobi symbol extends the notation of the Legendre symbol to this broader setting. Let $$ n>1 $$ be an odd integer with prime factorization $$ n=p_1^{e_1}p_2^{e_2}\cdots p_r^{e_r}. $$ For any integer $a$, the Jacobi symbol is defined by $$ \left(\frac{a}{n}\right)...

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Legendre Symbol

Definition Let $p$ be an odd prime and let $a\in\mathbb{Z}$. The Legendre symbol is defined by $$ \left(\frac{a}{p}\right) = \begin{cases} ;;;0 & \text{if } p\mid a,\ ;;;1 & \text{if } a \text{ is a quadratic residue modulo } p,\ -1 & \text{if } a \text{ is a quadratic nonresidue modulo } p. \end{cases} $$ genui{"math_block_widget_always_prefetch_v2":{"content":"\left(\frac{a}{p}\right)"} } Thus the Legendre symbol encodes whether the congruence $$ x^2\equiv a\pmod p $$ has...

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Squares Modulo $n$

Quadratic Congruences A quadratic congruence is a congruence involving a square. The basic form is $$ x^2\equiv a\pmod n, $$ where $a,n\in\mathbb{Z}$ and $n>1$. $$ x^2\equiv a\pmod n $$ The problem is to determine whether there exists an integer $x$ satisfying the congruence. If such an $x$ exists, then $a$ is called a quadratic residue modulo $n$. Otherwise, $a$ is called a quadratic nonresidue modulo $n$. Quadratic congruences form the...

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Appendix E. Topology Background

E.1 Motivation Topology studies continuity, convergence, connectedness, and geometric structure in an abstract setting. In number theory, topology appears naturally in real analysis, complex analysis, $p$-adic analysis, algebraic geometry, adelic theory, and automorphic forms. The real numbers carry a geometric topology derived from distance. The $p$-adic numbers carry a very different topology derived from divisibility. Modern number theory frequently moves between these worlds. Topology provides the language needed to describe...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Curves over Fields

Algebraic Curves An algebraic curve is a geometric object whose dimension is one. Curves are among the oldest and most important objects in number theory and algebraic geometry. Classically, a curve is defined by a polynomial equation in two variables: $$ f(x,y)=0. $$ Examples include: $$ y=x^2, $$ $$ x^2+y^2=1, $$ and $$ y^2=x^3-x. $$ Over a field $k$, such equations define subsets of affine or projective space whose coordinates...

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Finite Fields

Fields with Finitely Many Elements The familiar fields $$ \mathbb{Q},\quad \mathbb{R},\quad \mathbb{C} $$ contain infinitely many elements. In number theory and algebra, however, fields with only finitely many elements play an equally fundamental role. A finite field is a field containing finitely many elements. Such fields arise naturally in modular arithmetic, algebraic geometry, coding theory, and cryptography. The simplest examples come from arithmetic modulo a prime number. Prime Fields Let...

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Prime Counting Function

Counting Prime Numbers One of the oldest questions in number theory asks how prime numbers are distributed among the positive integers. Since primes become less frequent as numbers grow larger, mathematicians seek functions that measure their density. The central object is the prime counting function $$ \pi(x), $$ defined by $$ \pi(x)=#{p\leq x : p \text{ is prime}}. $$ Thus $\pi(x)$ counts the number of primes not exceeding $x$. For...

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Unique Prime Factorization

Statement of the Theorem The fundamental theorem of arithmetic states that every integer $n>1$ can be written as a product of prime numbers, and that this product is unique up to the order of the factors. Thus every integer $n>1$ has a representation $$ n=p_1p_2\cdots p_k, $$ where each $p_i$ is prime. The same factorization is often written by grouping equal primes: $$ n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}, $$ where $$ p_1<p_2<\cdots<p_r $$...

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Geometry of Diophantine Problems

From Equations to Shapes A Diophantine equation is first an arithmetic object. It asks for solutions in integers or rational numbers. But every polynomial equation also defines a geometric object. For example, $$ x^2+y^2=25 $$ defines a circle in the plane. The Diophantine problem asks for the lattice points on this circle, meaning the points $(x,y)$ with integer coordinates. Thus the same equation has two interpretations: Viewpoint Object of Study...

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Coprime Integers

Definition Two integers $a$ and $b$, not both zero, are called coprime if their greatest common divisor is $1$: $$ \gcd(a,b)=1. $$ They are also called relatively prime. Coprimality means that $a$ and $b$ have no common positive divisor except $1$. For example, $$ \gcd(8,15)=1, $$ so $8$ and $15$ are coprime. The integers themselves need not be prime. Both $8$ and $15$ are composite, but they share no prime...

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Bezout Identities

Linear Combinations Let $a$ and $b$ be integers. An integer of the form $$ ax+by $$ where $$ x,y\in\mathbb{Z} $$ is called an integer linear combination of $a$ and $b$. For example, if $a=18$ and $b=30$, then $$ 2a-b=2\cdot18-30=6 $$ is a linear combination of $18$ and $30$. Linear combinations are important because divisibility behaves well under addition and subtraction. If an integer $d$ divides both $a$ and $b$, then...

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Appendix D. Real and Complex Analysis Review

D.1 Real Numbers The real numbers $\mathbb{R}$ extend the rational numbers $\mathbb{Q}$ by filling gaps such as $$ \sqrt{2},\quad \pi,\quad e. $$ They form an ordered field. This means addition, multiplication, and order are compatible. For example, if $$ a<b, $$ then $$ a+c<b+c. $$ If $$ 0<c, $$ then $$ ac<bc. $$ The real numbers are also complete: every nonempty set of real numbers that is bounded above has...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Morphisms and Fibers

Maps Between Geometric Objects Geometry is not only concerned with spaces themselves, but also with maps between spaces. In algebraic geometry and arithmetic geometry, these maps are called morphisms. A morphism describes how one variety or scheme is transformed into another while preserving algebraic structure. For affine varieties, morphisms arise from polynomial functions. If $$ X\subseteq \mathbb{A}^n,\qquad Y\subseteq \mathbb{A}^m, $$ a morphism $$ f:X\to Y $$ is given by polynomial...

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Galois Groups

Symmetry of Polynomial Roots A polynomial equation may possess several roots related by hidden algebraic symmetries. Consider $$ f(x)=x^2-2. $$ Its roots are $$ \sqrt{2} \qquad \text{and} \qquad -\sqrt{2}. $$ These roots can be exchanged without affecting any rational number. Indeed, the transformation $$ \sqrt{2}\mapsto -\sqrt{2} $$ preserves all algebraic relations involving rational coefficients. For example, $$ (\sqrt{2})^2=2 $$ remains true after the substitution. Such transformations are called automorphisms of...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Abel Summation

Motivation In analytic number theory, one often studies sums of the form $$ \sum_{n\leq x} a_n f(n), $$ where $a_n$ is an arithmetic sequence and $f(x)$ is a smooth function. Direct estimation of such sums is frequently difficult. Abel summation transforms the problem into one involving partial sums of the coefficients $a_n$. This method is the discrete analogue of integration by parts. It is one of the basic tools connecting...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Rational and Integral Points

Polynomial Equations and Arithmetic Geometry A central problem in number theory is to study solutions of polynomial equations whose coordinates belong to a specified number system. Two important cases are: integral points, where the coordinates are integers, rational points, where the coordinates are rational numbers. For example, the equation $$ x^2+y^2=1 $$ has infinitely many rational solutions, such as $$ \left(\frac35,\frac45\right), $$ but only finitely many integral solutions: $$ (\pm1,0),\qquad...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Extended Euclidean Algorithm

From GCDs to Linear Combinations The Euclidean algorithm computes the greatest common divisor of two integers. The extended Euclidean algorithm does more. It also expresses the gcd as an integer linear combination of the original two integers. For integers $a$ and $b$, not both zero, it finds integers $x$ and $y$ such that $$ \gcd(a,b)=ax+by. $$ This identity is called Bézout's identity. For example, $$ \gcd(30,18)=6, $$ and $$ 6=30-18....

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Euclidean Algorithm

The Problem of Computing GCDs The greatest common divisor of two integers can be found by listing divisors, but this method becomes inefficient for large numbers. For example, finding $$ \gcd(1071,462) $$ by listing all divisors would be tedious. The Euclidean algorithm gives a fast and systematic method. It reduces a gcd problem to smaller gcd problems by repeated division with remainder. Basic Principle The key identity is $$ \gcd(a,b)=\gcd(b,r),...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Exponential Diophantine Equations

Equations Involving Powers An exponential Diophantine equation is a Diophantine equation in which one or more unknowns appear as exponents. Typical examples include $$ 2^x+3^y=z^2, $$ $$ x^n+y^n=z^n, $$ and $$ a^x-b^y=c. $$ Unlike linear or quadratic equations, exponential equations usually grow extremely rapidly. This rapid growth often implies that only finitely many integer solutions exist. Exponential Diophantine equations are among the most difficult problems in number theory because ordinary...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Catalan-Type Equations

Exponential Diophantine Equations A Catalan-type equation is a Diophantine equation involving powers whose values differ by a small amount. The classical example is $$ x^a-y^b=1, $$ where $$ x,y,a,b>1 $$ are integers. $$ x^a-y^b=1 $$ This equation asks whether two perfect powers can differ by exactly $1$. For example, $$ 3^2-2^3=9-8=1. $$ This gives the solution $$ (x,a,y,b)=(3,2,2,3). $$ A natural question arises: are there any others? Consecutive Perfect Powers...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Least Common Multiples

Common Multiples Let $a$ and $b$ be nonzero integers. An integer $m$ is called a common multiple of $a$ and $b$ if $$ a\mid m $$ and $$ b\mid m. $$ For example, the positive multiples of $6$ are $$ 6,12,18,24,30,36,\ldots $$ and the positive multiples of $8$ are $$ 8,16,24,32,40,48,\ldots $$ Their positive common multiples begin with $$ 24,48,72,\ldots $$ The smallest positive common multiple is $24$. Hence $$...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Convergence Methods

Infinite Processes in Analysis Analytic number theory studies infinite sums, products, and integrals. Before such expressions can be manipulated safely, one must understand the meaning of convergence. A sequence $(a_n)$ converges to $L$ if $$ \lim_{n\to\infty} a_n=L. $$ Similarly, an infinite series $$ \sum_{n=1}^{\infty} a_n $$ converges if its partial sums $$ S_N=\sum_{n=1}^{N} a_n $$ approach a finite limit as $N\to\infty$. Many techniques in analytic number theory depend on estimating...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Sums of Squares

Representation Problems One of the oldest questions in number theory asks which integers can be written as sums of squares. Typical examples are $$ 5=1^2+2^2, $$ $$ 25=3^2+4^2, $$ and $$ 50=1^2+7^2. $$ Such problems are called representation problems because they ask how integers may be represented by quadratic forms. The simplest equation of this type is $$ x^2+y^2=n, $$ where $n$ is a fixed positive integer and $x,y\in\mathbb{Z}$. $$...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Greatest Common Divisors

Common Divisors Let $a$ and $b$ be integers, not both zero. An integer $d$ is called a common divisor of $a$ and $b$ if $$ d\mid a $$ and $$ d\mid b. $$ For example, the positive divisors of $18$ are $$ 1,2,3,6,9,18, $$ and the positive divisors of $30$ are $$ 1,2,3,5,6,10,15,30. $$ Their common positive divisors are $$ 1,2,3,6. $$ The largest of these is $6$. Hence $$...

Writes › Book › Modern Number Theory › Appendix ›

Appendix C. Abstract Algebra Review

C.1 Algebraic Structures Abstract algebra studies sets equipped with operations. In number theory, these structures organize arithmetic behavior. The integers $\mathbb{Z}$ form a set with addition and multiplication. The residue classes modulo $n$, written $\mathbb{Z}/n\mathbb{Z}$, form a finite arithmetic system. Rational numbers, real numbers, complex numbers, finite fields, number fields, rings of integers, ideals, and elliptic curves all carry algebraic structure. The purpose of abstraction is to isolate the laws...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Euler Products

Series with Multiplicative Structure Euler products arise when an infinite series has coefficients controlled by multiplication. The simplest and most important example is the zeta series $$ \sum_{n=1}^{\infty}\frac{1}{n^s}, $$ where $s$ is a real number greater than $1$. This condition ensures absolute convergence. The key observation is that every positive integer factors uniquely into primes: $$ n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}. $$ Thus a sum over all positive integers can sometimes be decomposed...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Pell Equations

Quadratic Diophantine Equations A Pell equation is a Diophantine equation of the form $$ x^2-Dy^2=1, $$ where $D$ is a positive integer that is not a perfect square. $$ x^2-Dy^2=1 $$ The unknowns $x$ and $y$ are required to be integers. Equations of this type occupy a central position in classical number theory because they possess infinitely many solutions and exhibit deep algebraic structure. For example, when $$ D=2, $$...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

The Division Algorithm

Statement of the Algorithm The division algorithm is one of the basic structural facts about the integers. It says that any integer can be divided by a positive integer with a unique quotient and remainder. Let $a\in\mathbb{Z}$ and let $b\in\mathbb{N}$. Then there exist unique integers $q$ and $r$ such that $$ a=bq+r, $$ where $$ 0\le r<b. $$ The integer $q$ is called the quotient, and $r$ is called the...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Schemes

Motivation for Schemes Classical algebraic geometry studies varieties defined by polynomial equations. This theory works well over algebraically closed fields, especially over $\mathbb{C}$. However, many arithmetic questions require a more flexible framework. For example, number theory naturally involves rings such as $$ \mathbb{Z},\qquad \mathbb{Z}/n\mathbb{Z},\qquad \mathbb{Z}_p, $$ which are not fields. Classical varieties are insufficient for describing geometric structures over such rings. Scheme theory extends algebraic geometry from fields to arbitrary...

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Splitting Fields

Polynomials and Their Roots A central problem in algebra is to determine where a polynomial factors completely into linear terms. Consider the polynomial $$ f(x)=x^2-2. $$ Over the rational numbers $\mathbb{Q}$, this polynomial cannot be factored further, since $\sqrt{2}\notin\mathbb{Q}$. Over the real numbers $\mathbb{R}$, however, $$ x^2-2=(x-\sqrt{2})(x+\sqrt{2}). $$ Thus the factorization of a polynomial depends on the field in which it is considered. In general, a polynomial may fail to...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Infinite Products

From Sums to Products An infinite product has the form $$ \prod_{n=1}^{\infty} a_n = a_1a_2a_3\cdots, $$ where each $a_n$ is a real or complex number. As with infinite series, the meaning of the expression is defined by finite approximations. The $N$-th partial product is $$ P_N=\prod_{n=1}^{N}a_n. $$ The infinite product converges to $L$ if $$ \lim_{N\to\infty} P_N=L $$ exists and $L\neq 0$. The nonzero condition is important. If the product...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Pythagorean Triples

Right Triangles and Integer Solutions A Pythagorean triple is a triple of positive integers $$ (a,b,c) $$ satisfying the equation $$ a^2+b^2=c^2. $$ This equation arises from the Pythagorean theorem in Euclidean geometry. If a right triangle has legs of lengths $a$ and $b$, and hypotenuse $c$, then $$ a^2+b^2=c^2 $$ A classical example is $$ (3,4,5), $$ since $$ 3^2+4^2=9+16=25=5^2. $$ Other examples include $$ (5,12,13),\qquad (8,15,17),\qquad (7,24,25). $$...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Composite Numbers

Definition of Composite Numbers A positive integer $n>1$ is called composite if it is not prime. Equivalently, $n$ is composite if there exist integers $a$ and $b$ such that $$ n=ab, $$ with $$ 1<a<n, \qquad 1<b<n. $$ Thus a composite number can be factored into smaller positive integers. For example, $$ 4=2\cdot2, \qquad 6=2\cdot3, \qquad 12=3\cdot4. $$ Hence $4$, $6$, and $12$ are composite numbers. A composite integer therefore...

Writes › Book › Modern Number Theory › Appendix ›

Appendix B. Proof Techniques

B.1 The Nature of Mathematical Proof A mathematical proof is a logically complete argument establishing the truth of a statement from accepted assumptions, definitions, and previously proved results. In number theory, proofs often combine algebraic manipulation, divisibility arguments, congruences, induction, and structural reasoning. A proof must justify every essential step. Numerical examples may suggest a statement, but they do not establish it universally. The distinction between observation and proof is...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Prime Numbers

Definition of Prime Numbers Prime numbers are the fundamental building blocks of arithmetic. A positive integer $p>1$ is called prime if its only positive divisors are $$ 1 \quad\text{and}\quad p. $$ Thus a prime number cannot be factored into smaller positive integers. For example, $$ 2,3,5,7,11,13 $$ are prime numbers. The integer $2$ is prime because its positive divisors are only $$ 1 \quad\text{and}\quad 2. $$ Similarly, $13$ is prime...

Writes › Book › Modern Number Theory › Appendix ›

Appendix

A.1 Sets A set is a collection of objects, called its elements. If $x$ is an element of a set $A$, we write $x \in A$. If $x$ is not an element of $A$, we write $x \notin A$. Examples: $$ \mathbb{N}={1,2,3,\ldots} $$ $$ \mathbb{Z}={\ldots,-2,-1,0,1,2,\ldots} $$ $$ \mathbb{Q}=\left{\frac{a}{b}:a,b\in\mathbb{Z}, b\ne 0\right} $$ In number theory, sets usually describe domains of arithmetic objects: integers, primes, residue classes, ideals, rational points, or solutions...

Practice › LeetCode › LeetCode (archived) › LeetCode 08xx ›

LeetCode 866: Prime Palindrome

Problem Restatement We are given an integer n . We need to return the smallest integer greater than or equal to n that is both: Prime A palindrome A prime number has exactly two divisors: 1 and itself A palindrome reads the same from left to right and right to left. For example: 101 131 929 The answer is guaranteed to exist in the range: [2, 2 * 10^8] Input...

Practice › LeetCode › LeetCode (archived) › LeetCode 07xx ›

LeetCode 793: Preimage Size of Factorial Zeroes Function

Problem Restatement Define: f(x) as the number of trailing zeroes in: x! We are given an integer k . We need to return how many integers x satisfy: f(x) == k This count is called the size of the preimage of k . Input and Output Item Meaning Input Integer k Output Number of integers x such that f(x) == k f(x) Number of trailing zeroes in x! Important property...

Practice › LeetCode › LeetCode (archived) › LeetCode 09xx ›

LeetCode 914: X of a Kind in a Deck of Cards

Problem Restatement We are given an integer array deck . Each value represents the number written on one card. We need to decide whether we can split the entire deck into groups such that: Rule Meaning Same group size Every group has exactly x cards Minimum group size x > 1 Same card value All cards in one group have the same integer Return True if such a partition is...

Practice › LeetCode › LeetCode (archived) › LeetCode 08xx ›

LeetCode 858: Mirror Reflection

Problem Restatement We have a square room with mirrored walls. The side length of the room is p . A laser starts from the southwest corner. It first hits the east wall at a distance q from receptor 0 . There are three receptors: Receptor Corner 0 Southeast corner 1 Northeast corner 2 Northwest corner The southwest corner has no receptor. The laser reflects from the mirrors until it reaches...

Practice › LeetCode › LeetCode (archived) › LeetCode 06xx ›

LeetCode 633: Sum of Square Numbers

Problem Restatement We are given a non-negative integer c . We need to determine whether there exist two integers a and b such that: a * a + b * b == c If such integers exist, return True . Otherwise, return False . Input and Output Item Meaning Input A non-negative integer c Output True if c = a² + b² for some integers a and b , otherwise...

Practice › LeetCode › LeetCode (archived) › LeetCode 07xx ›

LeetCode 780: Reaching Points

Problem Restatement We are given four integers: sx, sy, tx, ty The starting point is: (sx, sy) The target point is: (tx, ty) From any point (x, y) , we may perform one of two moves: (x, y) -> (x + y, y) (x, y) -> (x, x + y) Return True if we can reach (tx, ty) from (sx, sy) . Otherwise, return False . Input and Output Item...

Practice › LeetCode › LeetCode (archived) › LeetCode 08xx ›

LeetCode 829: Consecutive Numbers Sum

Problem Restatement We are given a positive integer n . We need to count how many different ways n can be written as the sum of consecutive positive integers. For example: 15 = 15 15 = 7 + 8 15 = 4 + 5 + 6 15 = 1 + 2 + 3 + 4 + 5 So the answer for 15 is: 4 Input and Output Item Meaning Input...

Practice › LeetCode › LeetCode (archived) › LeetCode 08xx ›

LeetCode 878: Nth Magical Number

Problem Restatement A positive integer is called magical if it is divisible by either a or b . Given three integers n , a , and b , return the n th magical number. Because the answer can be very large, return it modulo: 10**9 + 7 The constraints are large: n can be up to 10^9 , while a and b can be up to 4 * 10^4 ....

Practice › LeetCode › LeetCode (archived) › LeetCode 05xx ›

LeetCode 507: Perfect Number

Problem Restatement A positive integer is called a perfect number if the sum of its positive divisors excluding itself equals the number. We are given an integer num . We need to determine whether num is a perfect number. The official problem defines a perfect number as a positive integer equal to the sum of all its positive divisors excluding itself. Input and Output Item Meaning Input An integer num...

Practice › LeetCode › LeetCode (archived) › LeetCode 09xx ›

LeetCode 952: Largest Component Size by Common Factor

Problem Restatement We are given an array nums of unique positive integers. Each number is a node in a graph. There is an undirected edge between two numbers if they share a common factor greater than 1 . Return the size of the largest connected component in this graph. For example, 6 and 15 are connected because they share factor 3 . The constraints are: Constraint Value nums.length 1 <=...

Writes › Book › Modern Number Theory › Chapter 5. Arithmetic Geometry and Modern Directions ›

Chapter 5. Arithmetic Geometry and Modern Directions

Polynomial Equations and Geometry Arithmetic geometry studies solutions of polynomial equations by combining algebra, geometry, and number theory. Its basic objects are spaces defined by polynomial equations. These spaces are called varieties. Let $k$ be a field, such as $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, or a finite field $\mathbb{F}_p$. The affine $n$-space over $k$ is $$ \mathbb{A}^n(k)=k^n. $$ Its points are ordered $n$-tuples $$ (a_1,\ldots,a_n), $$ where each $a_i\in k$. A polynomial...

Writes › Book › Modern Number Theory › Chapter 4. Algebraic Number Theory ›

Chapter 4. Algebraic Number Theory

Fields Inside Larger Fields A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, and the finite fields $\mathbb{F}_p$ are basic examples. A field extension occurs when one field is contained inside another. Definition. Let $K$ and $L$ be fields. We say that $L$ is a field extension of...

Writes › Book › Modern Number Theory › Chapter 3. Analytic Number Theory ›

Chapter 3. Analytic Number Theory

The Basic Series The harmonic series is the infinite series $$ 1+\frac12+\frac13+\frac14+\frac15+\cdots. $$ In summation notation, it is written as $$ \sum_{n=1}^{\infty}\frac1n. $$ This is one of the first infinite series that appears in analysis, but it is also central in number theory. Its terms become smaller and tend to zero: $$ \lim_{n\to\infty}\frac1n=0. $$ However, this condition alone does not guarantee that the infinite sum has a finite value. The...

Writes › Book › Modern Number Theory › Chapter 2. Classical Number Theory ›

Chapter 2. Classical Number Theory

Equations in Integers A Diophantine equation is an equation whose solutions are required to be integers. The unknowns are not allowed to range over the real numbers or complex numbers unless explicitly stated. This restriction changes the nature of the problem. For example, the equation $$ 2x+3y=1 $$ has many real solutions, but number theory asks for integer solutions. One such solution is $$ x=-1,\qquad y=1, $$ since $$ 2(-1)+3(1)=1....

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Divisibility Relations

Exact Division Division of integers does not always produce an integer. For example, $$ 12/3=4 $$ is an integer, while $$ 14/3 $$ is not. Number theory is mainly concerned with exact division, where one integer divides another without leaving a remainder. Let $a,b\in\mathbb{Z}$, with $a\ne0$. We say that $a$ divides $b$, and write $$ a\mid b, $$ if there exists an integer $q$ such that $$ b=aq. $$ The...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Historical Development of Number Systems

Numbers Before Mathematics The idea of number arose long before formal mathematics. Early civilizations used numbers for counting objects, measuring land, recording trade, and tracking time. The earliest number systems were concrete rather than abstract. A shepherd might count sheep using stones or marks on wood. Numbers represented physical collections rather than independent mathematical objects. Over time, arithmetic became increasingly symbolic and abstract. The development of number systems reflects the...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Growth of Integers

Size and Growth The integers extend infinitely in both directions: $$ \cdots,-3,-2,-1,0,1,2,3,\cdots $$ In number theory, one usually studies the size of an integer through its absolute value. The size of $n$ is measured by $$ |n|. $$ For positive integers, this is simply $n$. For negative integers, it removes the sign: $$ |-n|=n. $$ Growth concerns how quantities behave as $n$ becomes large. Many arithmetic questions are not about...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Recursive Definitions

Definitions by Recursion Many mathematical objects are defined recursively. A recursive definition specifies: one or more initial values, a rule for constructing later values from earlier ones. The natural numbers themselves arise recursively through the successor operation: $$ 1,2,3,\ldots $$ where each number is obtained from the preceding one by adding $1$. Recursive definitions are fundamental in number theory because arithmetic processes often proceed step by step. Sequences, algorithms, and...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Strong Induction

From Ordinary Induction to Strong Induction Ordinary induction proves a statement $P(n)$ by showing that truth passes from one case to the next: $$ P(k)\Longrightarrow P(k+1). $$ This is enough for many arguments. In number theory, however, the proof of $P(k+1)$ often requires more than the immediately preceding case. It may require several earlier cases, or even all earlier cases. Strong induction is designed for this situation. Instead of assuming...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Mathematical Induction

The Need for Induction Many statements in number theory concern all natural numbers. For example, one may wish to prove that $$ 1+2+\cdots+n=\frac{n(n+1)}{2} $$ for every $n\in\mathbb{N}$. It is not enough to check the formula for several values of $n$. Checking $$ n=1,2,3,4,5 $$ shows only that the formula holds in those cases. A proof must show that it holds for every natural number. Mathematical induction is the standard method...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Absolute Value and Distance

Magnitude and Sign The order relation distinguishes positive and negative integers, but in many situations the sign of a number is less important than its magnitude. For example, the integers $$ 5 \quad\text{and}\quad -5 $$ have opposite signs, yet both lie at the same distance from zero on the number line. This observation leads to the concept of absolute value. The absolute value of an integer $a$ is denoted by...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Order Relations

Ordered Structure of the Integers The integers are not merely a collection of numbers equipped with arithmetic operations. They also possess an order structure. Given two integers $a$ and $b$, one can determine whether $a$ is less than, equal to, or greater than $b$. The order relation on the integers is written using the symbols $$ <,\quad >,\quad \le,\quad \ge. $$ If $$ a<b, $$ we say that $a$ is...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Arithmetic Operations

Operations as Rules An arithmetic operation is a rule that combines numbers to produce another number. The most basic operations on integers are addition, subtraction, multiplication, and division. Addition and multiplication are always possible inside the integers. If $$ a,b\in\mathbb{Z}, $$ then $$ a+b\in\mathbb{Z} $$ and $$ ab\in\mathbb{Z}. $$ Subtraction is also always possible because the integers contain additive inverses: $$ a-b=a+(-b). $$ Division is different. The quotient $a/b$ need...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

The Integers

Limitations of the Natural Numbers The natural numbers are sufficient for counting and addition, but they are not sufficient for subtraction. For example, $$ 7-3=4 $$ is a natural number, while $$ 3-7 $$ is not contained in $\mathbb{N}$. To make subtraction universally possible, mathematics enlarges the number system by introducing the integers. The set of integers is denoted by $$ \mathbb{Z}={\ldots,-3,-2,-1,0,1,2,3,\ldots}. $$ The symbol $\mathbb{Z}$ comes from the German...

Writes › Book › Modern Number Theory › Chapter 1. Foundations of Arithmetic ›

Chapter 1. Foundations of Arithmetic

Counting and Quantity The natural numbers arise from the basic act of counting. When we count objects in a collection, we assign successive numbers: $$ 1,2,3,4,\ldots $$ These numbers are called the natural numbers. The standard notation for the set of natural numbers is $$ \mathbb{N}={1,2,3,4,\ldots}. $$ Some authors include $0$ in $\mathbb{N}$, while others begin with $1$. In this text, we shall usually write $$ \mathbb{N}_0={0,1,2,3,\ldots} $$ when zero...

Practice › LeetCode › LeetCode (archived) › LeetCode 03xx ›

LeetCode 365: Water and Jug Problem

Problem Restatement We are given two jugs with capacities x liters and y liters. There is an infinite water supply. We need to decide whether it is possible to measure exactly target liters using the two jugs. At the end, the total amount of water in both jugs must be exactly target . Allowed operations: Operation Meaning Fill Fill either jug completely Empty Empty either jug completely Pour Pour from...

Practice › LeetCode › LeetCode (archived) › LeetCode 02xx ›

LeetCode 263: Ugly Number

Problem Restatement We are given an integer n . Return True if n is an ugly number. An ugly number is a positive integer whose prime factors are limited to: 2, 3, 5 So a number is ugly if, after removing all factors of 2 , 3 , and 5 , nothing remains except 1 . The problem constraints are: -2^31 <= n <= 2^31 - 1 1 is considered...

Practice › LeetCode › LeetCode (archived) › LeetCode 02xx ›

LeetCode 258: Add Digits

Problem Restatement We are given a non-negative integer num . Repeatedly add all of its digits until the result has only one digit. Return that final one-digit number. Example: num = 38 First add its digits: 3 + 8 = 11 The result still has two digits, so add again: 1 + 1 = 2 Now the result has one digit. Answer: 2 The follow-up asks for an O(1) solution...

Practice › LeetCode › LeetCode (archived) › LeetCode 03xx ›

LeetCode 326: Power of Three

Problem Restatement We are given an integer n . Return true if n is a power of three. Otherwise, return false . A number is a power of three when it can be written as: n = 3^x where x is an integer. Examples: 27 = 3^3 9 = 3^2 3 = 3^1 1 = 3^0 So 27 , 9 , 3 , and 1 are powers of three. Numbers...

Maths ›

11. Number Theory

This volume studies integers and arithmetic structures. It covers divisibility, primes, modular arithmetic, Diophantine equations, and modern analytic and algebraic methods. Part I. Integers and Divisibility Chapter 1. Basic Properties of Integers 1.1 Divisibility and factors 1.2 Prime numbers 1.3 Greatest common divisor 1.4 Euclidean algorithm 1.5 Fundamental theorem of arithmetic Chapter 2. Arithmetic Functions 2.1 Definition and examples 2.2 Multiplicative functions 2.3 Möbius function 2.4 Euler totient function 2.5...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N7

IMO 2012 Shortlist N7 Category: Number Theory Problem Find all n ∈ N for which there exist nonnegative integers a1,a2,...,an such that 2a1 + 2a2 ··· + 2an = 3a1 3a2 ··· + n 3an = 1. Solution. Such numbers a1,a2,...,an exist if and only if n ≡ 1 (mod 4) or n ≡ 2 (mod 4). Let Pn k=1 k 3ak = 1 with a1,a2,...,an nonnegative integers. Then 1·x1+2·x2+···+n·xn...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N4

IMO 2017 Shortlist N4 Category: Number Theory Problem Call a rational number short if it has nitely many digits in its de imal expansion. For a positive integer m, we say that a positive integer t is m-tasti if there exists a number c P t1,2,3,...,2017u su h that 10t ´ 1 c ¨ m is short, and su h that 10k ´ 1 c ¨ m is not short...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N1

IMO 2017 Shortlist N1 Category: Number Theory Problem The sequen e a0,a1,a2,... of positive integers satises an 1 “ #? an, if ? an is an integer an 3, otherwise for every n ě 0. Determine all values of a0 ą 1 for whi h there is at least one number a su h that an “ a for innitely many values of n. (South Afri a)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N7

IMO 2021 Shortlist N7 Category: Number Theory Problem Let a1,a2,a3,... be an infinite sequence of positive integers such that an 2m divides an an m for allpositive integers n and m. Prove that this sequence is eventually periodic, i.e. there exist positive integers N and d such that an “an d for all n ąN.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist N4

IMO 2019 Shortlist N4 Category: Number Theory Problem Let Zą0 be the set of positive integers. A positive integer constant C is given. Find all functions f : Zą0 Ñ Zą0 such that, for all positive integers a and b satisfying a b ą C, a fpbq | a2 ` bfpaq. (Croatia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N2

IMO 2012 Shortlist N2 Category: Number Theory Problem Find all triples (x,y,z) of positive integers such that x ≤ y ≤ z and x3 (y3 z3 ) = 2012(xyz + 2). Solution. First note that x divides 2012·2 = 23 ·503. If 503 | x then the right-hand side of the equation is divisible by 5033 , and it follows that 5032 | xyz + 2. This is false as...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N6

IMO 2011 Shortlist N6 Category: Number Theory Problem Let P(x) and Q(x) be two polynomials with integer coefficients such that no nonconstant polynomial with rational coefficients divides both P(x) and Q(x). Suppose that for every positive integer n the integers P(n) and Q(n) are positive, and 2Q(n) − 1 divides 3P(n) − 1. Prove that Q(x) is a constant polynomial.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N1

IMO 2014 Shortlist N1 Category: Number Theory Problem Let n ě 2 be an integer, and let An be the set An “ t2n ´ 2k |k P Z, 0 ď k ă nu. Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of An. (Serbia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N2

IMO 2021 Shortlist N2 Category: Number Theory Problem Let n ě100 be an integer. The numbers n,n 1,...,2n are written on n 1 cards, one number per card. The cards are shuffled and divided into two piles. Prove that one of the piles contains two cards such that the sum of their numbers is a perfect square.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N7

IMO 2015 Shortlist N7 Category: Number Theory Problem Let Zą0 denote the set of positive integers. For any positive integer k, a function f : Zą0 Ñ Zą0 is called k-good if gcd fpmq n,fpnq ` m ˘ ď k for all m ‰ n. Find all k such that there exists a k-good function. (Canada)

Writes › Book › Modern Number Theory ›

Chapter 5. Arithmetic Geometry and Modern Directions

Section Title 1 Chapter 5. Arithmetic Geometry and Modern Directions 2 Schemes 3 Morphisms and Fibers 4 Curves over Fields 5 Arithmetic Surfaces 6 Étale Cohomology 7 Weil Conjectures 8 Representation Theory Background 9 Automorphic Representations 10 Adelic Methods 11 Langlands Program 12 Functoriality 13 Trace Formula 14 Fast Integer Arithmetic 15 Primality Testing 16 Integer Factorization 17 Lattice Reduction 18 Algorithms for Modular Forms 19 Algorithms for Elliptic Curves...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N4

IMO 2011 Shortlist N4 Category: Number Theory Problem For each positive integer k, let t(k) be the largest odd divisor of k. Determine all positive integers a for which there exists a positive integer n such that all the differences t(n + a) − t(n), t(n + a + 1) − t(n + 1), ..., t(n + 2a − 1) − t(n + a − 1) are divisible by 4.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist N5

IMO 2020 Shortlist N5 Category: Number Theory Problem Determine all functions f defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: piq fpnq ‰ 0 for at least one n; piiq fpxyq “ fpxq ` fpyq for every positive integers x and y; piiiq there are infinitely many positive integers n such that fpkq “ fpn ´ kq for all k ă...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N5

IMO 2023 Shortlist N5 Category: Number Theory Problem Let a1 ă a2 ă a3 ă ¨¨¨ be positive integers such that ak 1 divides 2pa1 a2 ¨¨¨ akq for every k ě 1. Suppose that for infinitely many primes p, there exists k such that p divides ak. Prove that for every positive integer n, there exists k such that n divides ak. (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N1

IMO 2021 Shortlist N1 Category: Number Theory Problem Determine all integers n ě 1 for which there exists a pair of positive integers pa,bq such that no cube of a prime divides a2 b 3 and ab 3b 8 a2 b 3 “n.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N1

IMO 2022 Shortlist N1 Category: Number Theory Problem A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.) (Cyprus)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist N2

IMO 2024 Shortlist N2 Category: Number Theory Problem Determine all finite, nonempty sets S of positive integers such that for every a, b P S there exists c P S with a | b ` 2c. (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N1

IMO 2007 Shortlist N1 Category: Number Theory Problem Find all pairs (k,n) of positive integers for which 7k − 3n divides k4 n2 . (Austria)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N8

IMO 2015 Shortlist N8 Category: Number Theory Problem For every positive integer n with prime factorization n “ śk i“1 pαi i , define ℧pnq “ ÿ i: pią10100 αi. That is, ℧pnq is the number of prime factors of n greater than 10100 , counted with multiplicity. Find all strictly increasing functions f : Z Ñ Z such that ℧ ` fpaq ´ fpbq ˘ ď ℧pa ´ bq...

Writes › Book › Modern Number Theory ›

Chapter 3. Analytic Number Theory

Section Title 1 Chapter 3. Analytic Number Theory 2 Infinite Products 3 Euler Products 4 Convergence Methods 5 Abel Summation 6 Prime Counting Function 7 Chebyshev Bounds 8 Prime Number Theorem 9 Logarithmic Integral 10 Error Terms 11 Short Intervals 12 Prime Gaps 13 Twin Prime Heuristics 14 Definition of the Zeta Function 15 Euler Product Formula 16 Analytic Continuation 17 Functional Equation 18 Zeros of the Zeta Function 19...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N1

IMO 2008 Shortlist N1 Category: Number Theory Problem Let n be a positive integer and let p be a prime number. Prove that if a, b, c are integers (not necessarily positive) satisfying the equations an pb = bn pc = cn pa, then a = b = c.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N1

IMO 2009 Shortlist N1 Category: Number Theory Problem A social club has n members. They have the membership numbers 1,2,...,n, respectively. From time to time members send presents to other members, including items they have already received as presents from other members. In order to avoid the embarrassing situation that a member might receive a present that he or she has sent to other members, the club adds the following...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N3

IMO 2016 Shortlist N3 Category: Number Theory Problem Define P(n) = n2 n + 1. For any positive integers a and b, the set {P(a),P(a + 1),P(a + 2),...,P(a + b)} is said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist N2

IMO 2006 Shortlist N2 Category: Number Theory Problem For x ∈ (0,1) let y ∈ (0,1) be the number whose nth digit after the decimal point is the (2n )th digit after the decimal point of x. Show that if x is rational then so is y. (Canada)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N8

IMO 2012 Shortlist N8 Category: Number Theory Problem Prove that for every prime p > 100 and every integer r there exist two integers a and b such that p divides a2 b5 − r. Solution 1. Throughout the solution, all congruence relations are meant modulo p. Fix p, and let P = {0,1,...,p−1} be the set of residue classes modulo p. For every r ∈ P, let Sr =...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N4

IMO 2008 Shortlist N4 Category: Number Theory Problem Let n be a positive integer. Show that the numbers 2n − 1 , 2n − 1 , 2n − 1 , ... , 2n − 1 2n−1 − 1 are congruent modulo 2n to 1,3,5,...,2n −1 in some order.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N8

IMO 2023 Shortlist N8 Category: Number Theory Problem Let Zą0 be the set of positive integers. Determine all functions f : Zą0 Ñ Zą0 such that fbfpaq pa 1q “ pa 1qfpbq holds for all a,b P Zą0, where fk pnq “ fpfp¨¨¨fpnq¨¨¨qq denotes the composition of f with itself k times. (Taiwan) 10 Chiba, Japan, 2nd–13th July 2023 Shortlisted problems – solutions 11

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N7

IMO 2013 Shortlist N7 Category: Number Theory Problem Let ν be an irrational positive number, and let m be a positive integer. A pair pa,bq of positive integers is called good if arbνs ´ btaνu “ m. A good pair pa,bq is called excellent if neither of the pairs pa´b,bq and pa,b´aq is good. (As usual, by txu and rxs we denote the integer numbers such that x ´ 1...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N1

IMO 2012 Shortlist N1 Category: Number Theory Problem Call admissible a set A of integers that has the following property: If x,y ∈ A (possibly x = y) then x2 kxy + y2 ∈ A for every integer k. Determine all pairs m,n of nonzero integers such that the only admissible set containing both m and n is the set of all integers. Solution. A pair of integers m,n fulfills...

Writes › Book › Modern Number Theory ›

Chapter 4. Algebraic Number Theory

Section Title 1 Chapter 4. Algebraic Number Theory 2 Splitting Fields 3 Galois Groups 4 Finite Fields 5 Cyclotomic Fields 6 Ramification 7 Absolute Values 8 $p$-Adic Numbers 9 Completion of Fields 10 Hensel’s Lemma 11 Local-Global Principles 12 Adeles and Ideles 13 Abelian Extensions 14 Reciprocity Maps 15 Hilbert Class Fields 16 Local Class Field Theory 17 Global Class Field Theory 18 Modular Groups 19 Modular Functions 20 Modular...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N2

IMO 2014 Shortlist N2 Category: Number Theory Problem Determine all pairs px,yq of positive integers such that a 7x2 ´ 13xy 7y2 “ |x ´ y| 1. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N8

IMO 2022 Shortlist N8 Category: Number Theory Problem Prove that 5n ´ 3n is not divisible by 2n ` 65 for any positive integer n. (Belgium) 8 Oslo, Norway, 6th–16th July 2022 Shortlisted problems – solutions 9

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N3

IMO 2007 Shortlist N3 Category: Number Theory Problem Let X be a set of 10000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset Y of X such that a − b + c − d + e is not divisible by 47 for any a,b,c,d,e ∈ Y . (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N3

IMO 2021 Shortlist N3 Category: Number Theory Problem Find all positive integers n with the following property: the k positive divisors of n have a permutation pd1,d2,...,dkq such that for every i “1,2,...,k, the number d1 ¨¨¨ di is a perfect square.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N3

IMO 2012 Shortlist N3 Category: Number Theory Problem Determine all integers m ≥ 2 such that every n with m ≤ n ≤ m divides the binomial coefficient n m−2n . Solution. The integers in question are all prime numbers. First we check that all primes satisfy the condition, and even a stronger one. Namely, if p is a prime then every n with 1 ≤ n ≤ p...

Writes › Book › Modern Number Theory ›

Chapter 1. Foundations of Arithmetic

Section Title 1 Chapter 1. Foundations of Arithmetic 2 The Integers 3 Arithmetic Operations 4 Order Relations 5 Absolute Value and Distance 6 Mathematical Induction 7 Strong Induction 8 Recursive Definitions 9 Growth of Integers 10 Historical Development of Number Systems 11 Divisibility Relations 12 Prime Numbers 13 Composite Numbers 14 The Division Algorithm 15 Greatest Common Divisors 16 Least Common Multiples 17 Euclidean Algorithm 18 Extended Euclidean Algorithm 19...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N4

IMO 2014 Shortlist N4 Category: Number Theory Problem Let n ą 1 be a given integer. Prove that infinitely many terms of the sequence pakqkě1, defined by ak “ Z nk k ^ , are odd. (For a real number x, txu denotes the largest integer not exceeding x.) (Hong Kong)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N8

IMO 2014 Shortlist N8 Category: Number Theory Problem For every real number x, let }x} denote the distance between x and the nearest integer. Prove that for every pair pa,bq of positive integers there exist an odd prime p and a positive integer k satisfying › › › › a pk › › › › › › › › b pk › › › › › › › › a...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist N7

IMO 2020 Shortlist N7 Category: Number Theory Problem Let S be a set consisting of n ě 3 positive integers, none of which is a sum of two other distinct members of S. Prove that the elements of S may be ordered as a1,a2,...,an so that ai does not divide ai´1 ai 1 for all i “ 2,3,...,n ´ 1. (Ukraine) Shortlisted problems 11 12 Saint-Petersburg — Russia, 18th–28th September...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N7

IMO 2014 Shortlist N7 Category: Number Theory Problem Let c ě 1 be an integer. Define a sequence of positive integers by a1 “ c and an 1 “ a3 n ´ 4c ¨ a2 n 5c2 ¨ an ` c for all n ě 1. Prove that for each integer n ě 2 there exists a prime number p dividing an but none of the numbers a1,...,an´1. (Austria)

Writes › Book › Modern Number Theory ›

Appendix

Section Title 1 Appendix 2 Appendix B. Proof Techniques 3 Appendix C. Abstract Algebra Review 4 Appendix D. Real and Complex Analysis Review 5 Appendix E. Topology Background 6 Appendix F. Measure and Integration 7 Appendix G. Linear Algebra Review 8 Appendix H. Category Theory Basics 9 Appendix I. Computational Tools 10 Appendix J. Historical Notes and Bibliography 11 Glossary 12 Notation Index 13 Index of Theorems 14 Index of...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N8

IMO 2011 Shortlist N8 Category: Number Theory Problem Let k be a positive integer and set n = 2k Prove that n is a prime number if and only if the following holds: there is a permutation a1,...,an−1 of the numbers 1,2,...,n − 1 and a sequence of integers g1,g2,...,gn−1 such that n divides gai i −ai+1 for every i ∈ {1,2,...,n−1}, where we set an = a1.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N3

IMO 2022 Shortlist N3 Category: Number Theory Problem Let a ą 1 be a positive integer, and let d ą 1 be a positive integer coprime to a. Let x1 “ 1 and, for k ě 1, define xk`1 “ xk ` d if a doesn’t divide xk, xk{a if a divides xk. Find the greatest positive integer n for which there exists an index k such that xk is...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist N4

IMO 2006 Shortlist N4 Category: Number Theory Problem Let P be a polynomial of degree n > 1 with integer coefficients and let k be any positive integer. Consider the polynomial Q(x) = P(P(...P(P(x))...)), with k pairs of parentheses. Prove that Q has no more than n integer fixed points, i.e. integers satisfying the equation Q(x) = x. (Romania)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N4

IMO 2009 Shortlist N4 Category: Number Theory Problem Find all positive integers n such that there exists a sequence of positive integers a1,a2,...,an satisfying ak+1 = a2 k + 1 ak−1 + 1 − 1 for every k with 2 ≤ k ≤ n − 1.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N6

IMO 2009 Shortlist N6 Category: Number Theory Problem Let k be a positive integer. Show that if there exists a sequence a0,a1,... of integers satisfying the condition an = an−1 + nk n for all n ≥ 1, then k − 2 is divisible by 3.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist N3

IMO 2019 Shortlist N3 Category: Number Theory Problem We say that a set S of integers is rootiful if, for any positive integer n and any a0,a1,...,an P S, all integer roots of the polynomial a0 a1x ¨¨¨ ` anxn are also in S. Find all rootiful sets of integers that contain all numbers of the form 2a ´ 2b for positive integers a and b. (Czech Republic)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist N5

IMO 2006 Shortlist N5 Category: Number Theory Problem Find all integer solutions of the equation x7 − 1 x − 1 = y5 − 1. (Russia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist N3

IMO 2010 Shortlist N3 Category: Number Theory Problem Find the smallest number n such that there exist polynomials f1, f2, ..., fn with rational coefficients satisfying x2 7 f1pxq2 f2pxq2 fnpxq2 . (Poland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N6

IMO 2014 Shortlist N6 Category: Number Theory Problem Let a1 ă a2 ă ¨¨¨ ă an be pairwise coprime positive integers with a1 being prime and a1 ě n ` 2. On the segment I “ r0,a1a2 ¨¨¨ans of the real line, mark all integers that are divisible by at least one of the numbers a1,...,an. These points split I into a number of smaller segments. Prove that the sum...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N5

IMO 2009 Shortlist N5 Category: Number Theory Problem Let P(x) be a non-constant polynomial with integer coefficients. Prove that there is no function T from the set of integers into the set of integers such that the number of integers x with Tn (x) = x is equal to P(n) for every n ≥ 1, where Tn denotes the n-fold application of T.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N2

IMO 2016 Shortlist N2 Category: Number Theory Problem Let τ(n) be the number of positive divisors of n. Let τ1(n) be the number of positive divisors of n which have remainders 1 when divided by 3. Find all possible integral values of the fraction τ(10n) τ1(10n) .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist N7

IMO 2019 Shortlist N7 Category: Number Theory Problem Prove that there is a constant c ą 0 and inﬁnitely many positive integers n with the following property: there are inﬁnitely many positive integers that cannot be expressed as the sum of fewer than cnlogpnq pairwise coprime nth powers. (Canada)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist N1

IMO 2010 Shortlist N1 Category: Number Theory Problem Find the least positive integer n for which there exists a set ts1,s2,...,snu consisting of n distinct positive integers such that 1 1 s1 1 1 s2 ... 1 1 sn 51 .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N8

IMO 2017 Shortlist N8 Category: Number Theory Problem Let p be an odd prime number and Zą0 be the set of positive integers. Suppose that a fun tion f : Zą0 ˆ Zą0 Ñ t0,1u satises the following properties: • fp1,1q “ 0; • fpa,bq fpb,aq “ 1 for any pair of relatively prime positive integers pa,bq not both equal to 1; • fpa b,bq “ fpa,bq for any pair...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist N2

IMO 2010 Shortlist N2 Category: Number Theory Problem Find all pairs pm,nq of nonnegative integers for which m2 2 3n m 2n 1 1 . (1) (Australia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N3

IMO 2015 Shortlist N3 Category: Number Theory Problem Let m and n be positive integers such that m ą n. Define xk “ pm kq{pn kq for k “ 1,2,...,n 1. Prove that if all the numbers x1,x2,...,xn 1 are integers, then x1x2 ¨¨¨xn`1 ´1 is divisible by an odd prime. (Austria)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N2

IMO 2009 Shortlist N2 Category: Number Theory Problem A positive integer N is called balanced, if N = 1 or if N can be written as a product of an even number of not necessarily distinct primes. Given positive integers a and b, consider the polynomial P defined by P(x) = (x + a)(x + b). (a) Prove that there exist distinct positive integers a and b such that all...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N5

IMO 2014 Shortlist N5 Category: Number Theory Problem Find all triples pp,x,yq consisting of a prime number p and two positive integers x and y such that xp´1 y and x yp´1 are both powers of p. (Belgium)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N6

IMO 2023 Shortlist N6 Category: Number Theory Problem A sequence of integers a0,a1,a2,... is called kawaii, if a0 “ 0,a1 “ 1, and, for any positive integer n, we have pan 1 ´ 3an 2an´1qpan 1 ´ 4an 3an´1q “ 0. An integer is called kawaii if it belongs to a kawaii sequence. Suppose that two consecutive positive integers m and m`1 are both kawaii (not necessarily belonging to the...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N7

IMO 2017 Shortlist N7 Category: Number Theory Problem Say that an ordered pair px,yq of integers is an irredu ible latti e point if x and y are relatively prime. For any nite set S of irredu ible latti e points, show that there is a homogenous polynomial in two variables, fpx,yq, with integer oe ients, of degree at least 1, su h that fpx,yq “ 1 for ea h...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist N4

IMO 2020 Shortlist N4 Category: Number Theory Problem For any odd prime p and any integer n, let dppnq P t0,1,...,p ´ 1u denote the remainder when n is divided by p. We say that pa0,a1,a2,...q is a p-sequence, if a0 is a positive integer coprime to p, and an 1 “ an dppanq for n ě 0. (a) Do there exist infinitely many primes p for which there exist...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N6

IMO 2007 Shortlist N6 Category: Number Theory Problem Let k be a positive integer. Prove that the number (4k2 − 1)2 has a positive divisor of the form 8kn − 1 if and only if k is even. (United Kingdom)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist N1

IMO 2019 Shortlist N1 Category: Number Theory Problem Find all pairs pm,nq of positive integers satisfying the equation p2n ´ 1qp2n ´ 2qp2n ´ 4q¨¨¨p2n ´ 2n´1 q “ m! (El Salvador)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N6

IMO 2021 Shortlist N6 Category: Number Theory Problem Determine all integers n ě 2 with the following property: every n pairwise distinct integers whose sum is not divisible by n can be arranged in some order a1, a2, ..., an so that n divides 1 ¨a1 2 ¨a2 ¨¨¨`n ¨an.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist N3

IMO 2014 Shortlist N3 Category: Number Theory Problem A coin is called a Cape Town coin if its value is 1{n for some positive integer n. Given a collection of Cape Town coins of total value at most 99 ` 1 , prove that it is possible to split this collection into at most 100 groups each of total value at most 1. (Luxembourg)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist N3

IMO 2024 Shortlist N3 Category: Number Theory Problem Determine all sequences a1, a2, ... of positive integers such that, for any pair of positive integers m ď n, the arithmetic and geometric means am am 1 ¨¨¨ an n ´ m 1 and pamam 1 ¨¨¨anq n´m`1 are both integers. (Singapore)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist N1

IMO 2020 Shortlist N1 Category: Number Theory Problem Given a positive integer k, show that there exists a prime p such that one can choose distinct integers a1,a2,...,ak 3 P t1,2,...,p ´ 1u such that p divides aiai 1ai 2ai 3 ´ i for all i “ 1,2,...,k. (South Africa)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N5

IMO 2017 Shortlist N5 Category: Number Theory Problem Find all pairs pp,qq of prime numbers with p ą q for whi h the number pp qqp q pp ´ qqp´q ´ 1 pp qqp´qpp ´ qqp q ´ 1 is an integer. (Japan) Shortlisted problems 11

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N3

IMO 2018 Shortlist N3 Category: Number Theory Problem Dene the sequen e a0,a1,a2,... by an “ 2n ` 2tn{2u . Prove that there are innitely many terms of the sequen e whi h an be expressed as a sum of (two or more) distin t terms of the sequen e, as well as innitely many of those whi h annot be expressed in su h a way. (Serbia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N7

IMO 2023 Shortlist N7 Category: Number Theory Problem Let a,b,c,d be positive integers satisfying ab a b cd c d “ pa bqpc dq a b c d . Determine all possible values of a b c ` d. (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N2

IMO 2011 Shortlist N2 Category: Number Theory Problem Consider a polynomial P(x) = (x + d1)(x + d2) · ... · (x + d9), where d1, d2,..., d9 are nine distinct integers. Prove that there exists an integer N such that for all integers x ≥ N the number P(x) is divisible by a prime number greater than 20.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist N4

IMO 2024 Shortlist N4 Category: Number Theory Problem Determine all positive integers a and b such that there exists a positive integer g such that gcdpan b,bn aq “ g for all sufficiently large n. (Indonesia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N4

IMO 2007 Shortlist N4 Category: Number Theory Problem For every integer k ≥ 2, prove that 23k divides the number 2k+1 2k − 2k 2k−1 (1) but 23k+1 does not. (Poland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N5

IMO 2016 Shortlist N5 Category: Number Theory Problem Let a be a positive integer which is not a square number. Denote by A the set of all positive integers k such that k = x2 − a x2 − y2 (1) for some integers x and y with x > √ a. Denote by B the set of all positive integers k such that (1) is satisfied for some integers...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N3

IMO 2013 Shortlist N3 Category: Number Theory Problem Prove that there exist infinitely many positive integers n such that the largest prime divisor of n4 n2 1 is equal to the largest prime divisor of pn 1q4 pn 1q2 1. (Belgium)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist N2

IMO 2019 Shortlist N2 Category: Number Theory Problem Find all triples pa,b,cq of positive integers such that a3 b3 c3 “ pabcq2 . (Nigeria)

Writes › Book › Modern Number Theory ›

Chapter 2. Classical Number Theory

Section Title 1 Chapter 2. Classical Number Theory 2 Pythagorean Triples 3 Pell Equations 4 Sums of Squares 5 Catalan-Type Equations 6 Exponential Diophantine Equations 7 Rational and Integral Points 8 Geometry of Diophantine Problems 9 Squares Modulo $n$ 10 Legendre Symbol 11 Jacobi Symbol 12 Euler Criterion 13 Quadratic Reciprocity 14 Gauss Sums 15 Higher Reciprocity Laws 16 Computational Aspects 17 Euclidean Algorithm Revisited 18 Finite Continued Fractions 19...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist N6

IMO 2006 Shortlist N6 Category: Number Theory Problem Let a > b > 1 be relatively prime positive integers. Define the weight of an integer c, denoted by w(c), to be the minimal possible value of |x| + |y| taken over all pairs of integers x and y such that ax + by = c. An integer c is called a local champion if w(c) ≥ w(c ± a) and...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist N7

IMO 2006 Shortlist N7 Category: Number Theory Problem Prove that, for every positive integer n, there exists an integer m such that 2m m is divisible by n. (Estonia)

Writes › Book ›

Modern Number Theory

Modern Number Theory Chapter Title Description 1 Chapter 1. Foundations of Arithmetic The natural numbers arise from the basic act of counting. When we count objects in a collection, we assign successive numbers: 2 Chapter 2. Classical Number Theory A Diophantine equation is an equation whose solutions are required to be integers. The unknowns are not allowed to range over the real numbers or complex numbers unless... 3 Chapter 3....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N6

IMO 2018 Shortlist N6 Category: Number Theory Problem Let f : t1,2,3,...u Ñ t2,3,...u be a fun tion su h that fpm nq | fpmq fpnq for all pairs m,n of positive integers. Prove that there exists a positive integer c ą 1 whi h divides all values of f. (Mexi o)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N3

IMO 2023 Shortlist N3 Category: Number Theory Problem For positive integers n and k ě 2 define Ekpnq as the greatest exponent r such that kr divides n!. Prove that there are infinitely many n such that E10pnq ą E9pnq and infinitely many m such that E10pmq ă E9pmq. (Brazil)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N6

IMO 2015 Shortlist N6 Category: Number Theory Problem Let Zą0 denote the set of positive integers. Consider a function f : Zą0 Ñ Zą0. For any m,n P Zą0 we write fn pmq “ fpfp...f looomooon n pmq...qq. Suppose that f has the following two properties: piq If m,n P Zą0, then fn pmq ´ m n P Zą0; piiq The set Zą0 z tfpnq|n P Zą0u is finite. Prove...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N1

IMO 2016 Shortlist N1 Category: Number Theory Problem For any positive integer k, denote the sum of digits of k in its decimal representation by S(k). Find all polynomials P(x) with integer coefficients such that for any positive integer n ⩾ 2016, the integer P(n) is positive and S(P(n)) = P(S(n)).

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N4

IMO 2015 Shortlist N4 Category: Number Theory Problem Suppose that a0,a1,... and b0,b1,... are two sequences of positive integers satisfying a0,b0 ě 2 and an 1 “ gcdpan,bnq 1, bn 1 “ lcmpan,bnq ´ 1 for all n ě 0. Prove that the sequence (an) is eventually periodic; in other words, there exist integers N ě 0 and t ą 0 such that an t “ an for all n...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N2

IMO 2023 Shortlist N2 Category: Number Theory Problem Determine all pairs pa,pq of positive integers with p prime such that pa `a4 is a perfect square. (Bangladesh)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N8

IMO 2021 Shortlist N8 Category: Number Theory Problem For a polynomial Ppxq with integer coefficients let P1 pxq “ Ppxq and Pk`1 pxq “ PpPk pxqq for k ě1. Find all positive integers n for which there exists a polynomial Ppxq with integer coefficients such that for every integer m ě 1, the numbers Pm p1q,...,Pm pnq leave exactly rn{2m s distinct remainders when divided by n. Shortlisted problems –...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist N6

IMO 2010 Shortlist N6 Category: Number Theory Problem The rows and columns of a 2n 2n table are numbered from 0 to 2n 1. The cells of the table have been colored with the following property being satisfied: for each 0 ¨i,j ¨2n 1, the jth cell in the ith row and the pi jqth cell in the jth row have the same color. (The indices of the cells in...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N5

IMO 2018 Shortlist N5 Category: Number Theory Problem Four positive integers x, y, z, and t satisfy the relations xy ´ zt “ x y “ z t. Is it possible that both xy and zt are perfe t squares? (Russia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N1

IMO 2011 Shortlist N1 Category: Number Theory Problem For any integer d > 0, let f(d) be the smallest positive integer that has exactly d positive divisors (so for example we have f(1) = 1, f(5) = 16, and f(6) = 12). Prove that for every integer k ≥ 0 the number f(2k ) divides f(2k+1 ).

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N1

IMO 2013 Shortlist N1 Category: Number Theory Problem Let Zą0 be the set of positive integers. Find all functions f : Zą0 Ñ Zą0 such that m2 fpnq | mfpmq n for all positive integers m and n. (Malaysia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N2

IMO 2007 Shortlist N2 Category: Number Theory Problem Let b,n > 1 be integers. Suppose that for each k > 1 there exists an integer ak such that b − an k is divisible by k. Prove that b = An for some integer A. (Canada)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N5

IMO 2007 Shortlist N5 Category: Number Theory Problem Find all surjective functions f : N → N such that for every m,n ∈ N and every prime p, the number f(m + n) is divisible by p if and only if f(m) + f(n) is divisible by p. (N is the set of all positive integers.) (Iran)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N6

IMO 2013 Shortlist N6 Category: Number Theory Problem Determine all functions f : Q ÝÑ Z satisfying f ˆ fpxq a b ˙ “ f ´x a b ¯ for all x P Q, a P Z, and b P Zą0. (Here, Zą0 denotes the set of positive integers.) (Israel)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N5

IMO 2011 Shortlist N5 Category: Number Theory Problem Let f be a function from the set of integers to the set of positive integers. Suppose that for any two integers m and n, the difference f(m) − f(n) is divisible by f(m − n). Prove that for all integers m, n with f(m) ≤ f(n) the number f(n) is divisible by f(m).

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N6

IMO 2016 Shortlist N6 Category: Number Theory Problem Denote by N the set of all positive integers. Find all functions f : N → N such that for all positive integers m and n, the integer f(m) + f(n) − mn is nonzero and divides mf(m) + nf(n). 10 IMO 2016 Hong Kong

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N7

IMO 2011 Shortlist N7 Category: Number Theory Problem Let p be an odd prime number. For every integer a, define the number Sa = a + a2 ··· + ap−1 p − 1 . Let m and n be integers such that S3 + S4 − 3S2 = m n . Prove that p divides m.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N1

IMO 2018 Shortlist N1 Category: Number Theory Problem Determine all pairs pn,kq of distin t positive integers su h that there exists a positive integer s for whi h the numbers of divisors of sn and of sk are equal. (Ukraine)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N6

IMO 2008 Shortlist N6 Category: Number Theory Problem Prove that there exist infinitely many positive integers n such that n2 1 has a prime divisor greater than 2n + √ 2n.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N5

IMO 2008 Shortlist N5 Category: Number Theory Problem For every n ∈ N let d(n) denote the number of (positive) divisors of n. Find all func- tions f : N → N with the following properties: (i) d(f(x)) = x for all x ∈ N; (ii) f(xy) divides (x − 1)yxy−1 f(x) for all x,y ∈ N.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N4

IMO 2021 Shortlist N4 Category: Number Theory Problem Alice is given a rational number r ą 1 and a line with two points B ‰ R, where point R contains a red bead and point B contains a blue bead. Alice plays a solitaire game by performinga sequence of moves. In every move, she chooses a (not necessarily positive) integer k, and a bead to move. If that bead is...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist N5

IMO 2024 Shortlist N5 Category: Number Theory Problem Let S be a finite nonempty set of prime numbers. Let 1 “ b1 ă b2 ă ¨¨¨ be the sequence of all positive integers whose prime divisors all belong to S. Prove that, for all but finitely many positive integers n, there exist positive integers a1, a2, ..., an such that a1 b1 a2 b2 ¨¨¨ an bn “ R b1...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N6

IMO 2022 Shortlist N6 Category: Number Theory Problem Let Q be a set of prime numbers, not necessarily finite. For a positive integer n consider its prime factorisation; define ppnq to be the sum of all the exponents and qpnq to be the sum of the exponents corresponding only to primes in Q. A positive integer n is called special if ppnq ppn 1q and qpnq qpn 1q are both...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist N6

IMO 2020 Shortlist N6 Category: Number Theory Problem For a positive integer n, let dpnq be the number of positive divisors of n, and let ϕpnq be the number of positive integers not exceeding n which are coprime to n. Does there exist a constant C such that ϕpdpnqq dpϕpnqq ď C for all n ě 1? (Cyprus)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N4

IMO 2016 Shortlist N4 Category: Number Theory Problem Let n,m,k and l be positive integers with n 6= 1 such that nk +mnl +1 divides nk+l −1. Prove that • m = 1 and l = 2k; or • l|k and m = nk−l−1 nl−1 .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N2

IMO 2017 Shortlist N2 Category: Number Theory Problem Let p ě 2 be a prime number. Eduardo and Fernando play the following game making moves alternately: in ea h move, the urrent player hooses an index i in the set t0,1,...,p´1u that was not hosen before by either of the two players and then hooses an element ai of the set t0,1,2,3,4,5,6,7,8,9u. Eduardo has the rst move. The game ends...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist N2

IMO 2020 Shortlist N2 Category: Number Theory Problem For each prime p, there is a kingdom of p-Landia consisting of p islands numbered 1, 2, ..., p. Two distinct islands numbered n and m are connected by a bridge if and only if p divides pn2 ´ m 1qpm2 ´ n 1q. The bridges may pass over each other, but cannot cross. Prove that for infinitely many p there are...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N7

IMO 2009 Shortlist N7 Category: Number Theory Problem Let a and b be distinct integers greater than 1. Prove that there exists a positive integer n such that (an − 1)(bn − 1) is not a perfect square.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist N4

IMO 2010 Shortlist N4 Category: Number Theory Problem Let a,b be integers, and let Ppxqax3 bx. For any positive integer n we say that the pair pa,bq is n-good if n PpmqPpkq implies n m k for all integers m,k. We say that pa,bq is very good if pa,bq is n-good for infinitely many positive integers n. (a) Find a pair pa,bq which is 51-good, but...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist N1

IMO 2024 Shortlist N1 Category: Number Theory Problem Find all positive integers n with the following property: for all positive divisors d of n, we have that d 1 | n or d 1 is prime. (Ghana)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N6

IMO 2012 Shortlist N6 Category: Number Theory Problem Let x and y be positive integers. If x2n − 1 is divisible by 2n y + 1 for every positive integer n, prove that x = 1. Solution. First we prove the following fact: For every positive integer y there exist infinitely many primes p ≡ 3 (mod 4) such that p divides some number of the form 2n y +...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N7

IMO 2007 Shortlist N7 Category: Number Theory Problem For a prime p and a positive integer n, denote by νp(n) the exponent of p in the prime factorization of n!. Given a positive integer d and a finite set {p1,...,pk} of primes. Show that there are infinitely many positive integers n such that d νpi (n) for all 1 ≤ i ≤ k. (India)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N2

IMO 2018 Shortlist N2 Category: Number Theory Problem Let n ą 1 be a positive integer. Ea h ell of an n ˆ n table ontains an integer. Suppose that the following onditions are satised: piq Ea h number in the table is ongruent to 1 modulo n; piiq The sum of numbers in any row, as well as the sum of numbers in any olumn, is ongruent to n...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N5

IMO 2021 Shortlist N5 Category: Number Theory Problem Prove that there are only finitely many quadruples pa,b,c,nq of positive integers such that n! “an´1 bn´1 cn´1 .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N4

IMO 2018 Shortlist N4 Category: Number Theory Problem Let a1, a2, ..., an, ... be a sequen e of positive integers su h that a1 a2 a2 a3 ¨¨¨ an´1 an an a1 is an integer for all n ě k, where k is some positive integer. Prove that there exists a positive integer m su h that an “ an`1 for all n ě m. (Mongolia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N1

IMO 2015 Shortlist N1 Category: Number Theory Problem Determine all positive integers M for which the sequence a0,a1,a2,..., defined by a0 “ 2M 1 and ak 1 “ aktaku for k “ 0,1,2,..., contains at least one integer term. (Luxembourg)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist N6

IMO 2019 Shortlist N6 Category: Number Theory Problem Let H “ X i ? : i P Zą0 ( “ t1,2,4,5,7,...u, and let n be a positive integer. Prove that there exists a constant C such that, if A Ă t1,2,...,nu satisﬁes |A| ě C ? n, then there exist a,b P A such that a´b P H. (Here Zą0 is the set of positive integers, and tzu denotes the...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N5

IMO 2015 Shortlist N5 Category: Number Theory Problem Determine all triples pa,b,cq of positive integers for which ab ´ c, bc ´ a, and ca ´ b are powers of 2. Explanation: A power of 2 is an integer of the form 2n , where n denotes some nonnegative integer. (Serbia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N4

IMO 2012 Shortlist N4 Category: Number Theory Problem An integer a is called friendly if the equation (m2 n)(n2 m) = a(m − n)3 has a solution over the positive integers. a) Prove that there are at least 500 friendly integers in the set {1,2,...,2012}. b) Decide whether a = 2 is friendly. Solution. a) Every a of the form a = 4k − 3 with k ≥ 2 is...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N2

IMO 2013 Shortlist N2 Category: Number Theory Problem Prove that for any pair of positive integers k and n there exist k positive integers m1,m2,...,mk such that 1 2k ´ 1 n “ ˆ 1 m1 ˙ˆ 1 m2 ˙ ¨¨¨ ˆ 1 mk ˙ . (Japan)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist N3

IMO 2006 Shortlist N3 Category: Number Theory Problem The sequence f(1),f(2),f(3),... is defined by f(n) = n jn k + jn k ··· + jn n k , where bxc denotes the integer part of x. (a) Prove that f(n + 1) > f(n) infinitely often. (b) Prove that f(n + 1) < f(n) infinitely often. (South Africa)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N5

IMO 2022 Shortlist N5 Category: Number Theory Problem For each 1 ď i ď 9 and T P N, define dipTq to be the total number of times the digit i appears when all the multiples of 1829 between 1 and T inclusive are written out in base 10. Show that there are infinitely many T P N such that there are precisely two distinct values among d1pTq,d2pTq,...,d9pTq. (United Kingdom)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N4

IMO 2023 Shortlist N4 Category: Number Theory Problem Let a1,a2,...,an,b1,b2,...,bn be 2n positive integers such that the n ` 1 products a1a2a3 ¨¨¨an, b1a2a3 ¨¨¨an, b1b2a3 ¨¨¨an, . . . b1b2b3 ¨¨¨bn form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression. (Canada)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N4

IMO 2013 Shortlist N4 Category: Number Theory Problem Determine whether there exists an infinite sequence of nonzero digits a1,a2,a3,... and a positive integer N such that for every integer k ą N, the number akak´1 ...a1 is a perfect square. (Iran)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist N2

IMO 2008 Shortlist N2 Category: Number Theory Problem Let a1,a2,...,an be distinct positive integers, n ≥ 3. Prove that there exist distinct indices i and j such that ai + aj does not divide any of the numbers 3a1,3a2,...,3an.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N3

IMO 2009 Shortlist N3 Category: Number Theory Problem Let f be a non-constant function from the set of positive integers into the set of positive integers, such that a − b divides f(a) − f(b) for all distinct positive integers a,b. Prove that there exist infinitely many primes p such that p divides f(c) for some positive integer c.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist N5

IMO 2010 Shortlist N5 Category: Number Theory Problem Let N be the set of all positive integers. Find all functions f : N Ñ N such that the number fpmq n m fpnq is a square for all m,n PN. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist N6

IMO 2024 Shortlist N6 Category: Number Theory Problem Let n be a positive integer. We say that a polynomial P with integer coeffi- cients is n-good if there exists a polynomial Q of degree 2 with integer coefficients such that QpkqpPpkq ` Qpkqq is never divisible by n for any integer k. Determine all integers n such that every polynomial with integer coefficients is an n-good polynomial. (France)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N7

IMO 2022 Shortlist N7 Category: Number Theory Problem Let k be a positive integer and let S be a finite set of odd prime numbers. Prove that there is at most one way (modulo rotation and reflection) to place the elements of S around a circle such that the product of any two neighbors is of the form x2 x k for some positive integer x. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist N8

IMO 2019 Shortlist N8 Category: Number Theory Problem Let a and b be two positive integers. Prove that the integer a2 ` R 4a2 b V is not a square. (Here rzs denotes the least integer greater than or equal to z.) (Russia) 10 Bath — UK, 11th–22nd July 2019

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist N8

IMO 2016 Shortlist N8 Category: Number Theory Problem Find all polynomials P(x) of odd degree d and with integer coefficients satisfying the following property: for each positive integer n, there exist n positive integers x1,x2,...,xn such that 1 < P(xi) P(xj) < 2 and P(xi) P(xj) is the d-th power of a rational number for every pair of indices i and j with 1 ⩽ i,j ⩽ n. Shortlisted problems...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist N5

IMO 2019 Shortlist N5 Category: Number Theory Problem Let a be a positive integer. We say that a positive integer b is a-good if an b ˘ ´ 1 is divisible by an 1 for all positive integers n with an ě b. Suppose b is a positive integer such that b is a-good, but b 2 is not a-good. Prove that b 1 is prime. (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N4

IMO 2022 Shortlist N4 Category: Number Theory Problem Find all triples of positive integers pa,b,pq with p prime and ap “ b! ` p. (Belgium)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N5

IMO 2012 Shortlist N5 Category: Number Theory Problem For a nonnegative integer n define rad(n) = 1 if n = 0 or n = 1, and rad(n) = p1p2 ···pk where p1 < p2 < ··· < pk are all prime factors of n. Find all polynomials f(x) with nonnegative integer coefficients such that rad(f(n)) divides rad(f(nrad(n) )) for every nonnegative integer n. Solution 1. We are going to prove...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist N11

IMO 2010 Shortlist N11 Category: Number Theory Problem . Same as Problem N1, but the constant is replaced by . (Canada) Answer for Problem N1. n 39. Solution for Problem N1. Suppose that for some n there exist the desired numbers; we may assume that s1 s2 sn. Surely s1 ¡ 1 since otherwise 1 1 s1 0. So we have 2 ¨s1 ¨s2 1 ¨¨sn pn...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist N1

IMO 2023 Shortlist N1 Category: Number Theory Problem Determine all positive, composite integers n that satisfy the following property: if the positive divisors of n are 1 “ d1 ă d2 ă ¨¨¨ ă dk “ n, then di divides di 1 di`2 for every 1 ď i ď k ´ 2. (Colombia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist N1

IMO 2006 Shortlist N1 Category: Number Theory Problem Determine all pairs (x,y) of integers satisfying the equation 1 + 2x 22x+1 = y2 . (USA)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist N2

IMO 2022 Shortlist N2 Category: Number Theory Problem Find all positive integers n ą 2 such that n! ˇ ˇ ˇ ˇ ź păqďn, p,q primes pp ` qq. (Nigeria)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N3

IMO 2011 Shortlist N3 Category: Number Theory Problem Let n ≥ 1 be an odd integer. Determine all functions f from the set of integers to itself such that for all integers x and y the difference f(x) − f(y) divides xn − yn .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist N5

IMO 2013 Shortlist N5 Category: Number Theory Problem Fix an integer k ě 2. Two players, called Ana and Banana, play the following game of numbers: Initially, some integer n ě k gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number m just written on the blackboard and replaces it by some number m1 with k ď...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2015 Shortlist ›

IMO 2015 Shortlist N2

IMO 2015 Shortlist N2 Category: Number Theory Problem Let a and b be positive integers such that a!b! is a multiple of a! b!. Prove that 3a ě 2b 2. (United Kingdom)

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IMO 2008 Shortlist N3

IMO 2008 Shortlist N3 Category: Number Theory Problem Let a0,a1,a2,... be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, gcd(ai,ai+1) > ai−1. Prove that an ≥ 2n for all n ≥ 0.

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IMO 2016 Shortlist N7

IMO 2016 Shortlist N7 Category: Number Theory Problem Let n be an odd positive integer. In the Cartesian plane, a cyclic polygon P with area S is chosen. All its vertices have integral coordinates, and all squares of its side lengths are divisible by n. Prove that 2S is an integer divisible by n.

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IMO 2017 Shortlist N6

IMO 2017 Shortlist N6 Category: Number Theory Problem Find the smallest positive integer n, or show that no su h n exists, with the following property: there are innitely many distin t n-tuples of positive rational numbers pa1,a2,...,anq su h that both a1 a2 ¨¨¨ an and a1 a2 ¨¨¨ an are integers. (Singapore)

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IMO 2024 Shortlist N7

IMO 2024 Shortlist N7 Category: Number Theory Problem Let Zą0 denote the set of positive integers. Let f : Zą0 Ñ Zą0 be a function satisfying the following property: for m, n P Zą0, the equation fpmnq2 “ fpm2 qfpfpnqqfpmfpnqq holds if and only if m and n are coprime. For each positive integer n, determine all the possible values of fpnq. (Japan) 10 Bath, United Kingdom, 10th –22nd July...

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IMO 2020 Shortlist N3

IMO 2020 Shortlist N3 Category: Number Theory Problem Let n be an integer with n ě 2. Does there exist a sequence pa1,...,anq of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N7

IMO 2018 Shortlist N7 Category: Number Theory Problem Let n ě 2018 be an integer, and let a1,a2,...,an,b1,b2,...,bn be pairwise distin t positive integers not ex eeding 5n. Suppose that the sequen e a1 b1 , a2 b2 , ..., an bn forms an arithmeti progression. Prove that the terms of the sequen e are equal. (Thailand) 8 Cluj-Napo a Romania, 314 July 2018 Shortlisted problems solutions 9

Practice › Mathematics › IMO › IMO Shortlist › IMO 2017 Shortlist ›

IMO 2017 Shortlist N3

IMO 2017 Shortlist N3 Category: Number Theory Problem Determine all integers n ě 2 with the following property: for any integers a1,a2,...,an whose sum is not divisible by n, there exists an index 1 ď i ď n su h that none of the numbers ai,ai ai 1,...,ai ai 1 ¨¨¨ ai`n´1 is divisible by n. (We let ai “ ai´n when i ą n.) (Thailand)

#number-theory

April Fools Day Contest

CF 171F - ucyhf

CF 2218D - The 67th OEIS Problem

Codeforces Round 1093 (Div. 1)

CF 2219A - Grid L

18.9 Factorization

18.22 Practical Number Theory Patterns

18.14 Möbius Function and Möbius Inversion

18.23 Complexity Analysis

18.2 Euclidean Algorithm

18.4 Modular Arithmetic

18.10 Chinese Remainder Theorem

18.5 Modular Inverse

18.13 Modular Combinatorics

18.20 Pollard's Rho Factorization

18.1 Divisibility

18.6 Fast Exponentiation

18.25 Number Theory Toolkit

18.7 Primality Testing

18.17 Primitive Roots

18.19 Miller-Rabin Primality Test

18.18 Discrete Logarithms

18.12 Euler's Theorem and Fermat's Little Theorem

Chapter 18. Number Theory

18.21 Big Integer Concerns

18.11 Euler Phi Function

18.3 Extended Euclidean Algorithm

18.24 Testing Number-Theoretic Code

18.15 Linear Congruences

18.8 Sieve of Eratosthenes

18.16 Diophantine Equations

CF 2226B - Everything Everywhere

CF 2226F - Inversion Invasion

Codeforces Round 1095 (Div. 2)

LeetCode 3671 - Sum of Beautiful Subsequences

LeetCode 3669 - Balanced K-Factor Decomposition

LeetCode 3610 - Minimum Number of Primes to Sum to Target

CF 1950E - Nearly Shortest Repeating Substring

Codeforces Round 943 (Div. 3)

CF 1954C - Long Multiplication

CF 1965B - Missing Subsequence Sum

Codeforces Round 935 (Div. 3)

CF 1973F - Maximum GCD Sum Queries

Codeforces Round 279 (Div. 2)

CF 1950D - Product of Binary Decimals

CF 1967B2 - Reverse Card (Hard Version)

CF 495B - Modular Equations

CF 487C - Prefix Product Sequence

Codeforces Round 282 (Div. 2)

Codeforces Round 275 (Div. 2)

CF 490D - Chocolate

CF 1955G - GCD on a grid

CF 1968A - Maximize?

CF 490C - Hacking Cypher

CF 1954E - Chain Reaction

CF 1945B - Fireworks

CF 1967B1 - Reverse Card (Easy Version)

Codeforces Round 937 (Div. 4)

Codeforces Round 278 (Div. 1)

CF 483A - Counterexample

Educational Codeforces Round 164 (Rated for Div. 2)

CF 1945H - GCD is Greater

LeetCode 3629 - Minimum Jumps to Reach End via Prime Teleportation

LeetCode 3867 - Sum of GCD of Formed Pairs

LeetCode 3855 - Sum of K-Digit Numbers in a Range

LeetCode 3850 - Count Sequences to K

LeetCode 3725 - Count Ways to Choose Coprime Integers from Rows

LeetCode 3765 - Complete Prime Number

LeetCode 3569 - Maximize Count of Distinct Primes After Split

LeetCode 3556 - Sum of Largest Prime Substrings

LeetCode 3770 - Largest Prime from Consecutive Prime Sum

LeetCode 3715 - Sum of Perfect Square Ancestors

LeetCode 3658 - GCD of Odd and Even Sums

LeetCode 3618 - Split Array by Prime Indices

LeetCode 3605 - Minimum Stability Factor of Array

LeetCode 3591 - Check if Any Element Has Prime Frequency

LeetCode 3589 - Count Prime-Gap Balanced Subarrays

LeetCode 3574 - Maximize Subarray GCD Score

LeetCode 3461 - Check If Digits Are Equal in String After Operations I