Prime factorization

Definition

Let $n$ be a positive integer. A prime factorization of $n$ is a finite list of prime numbers $p_1 \le p_2 \le \dots \le p_k$ such that $n = p_1 p_2 \cdots p_k$ . For $n = 1$ , the prime factorization is the empty list, with the empty product $1$ .

Remarks

We often group equal primes and write $n = q_1^{e_1} q_2^{e_2} \cdots q_r^{e_r}$ , with distinct primes $q_1 < \dots < q_r$ and exponents $e_i \ge 1$ . For example, the prime factorization of $360$ is $2, 2, 2, 3, 3, 5$ , that is, $360 = 2^3 \cdot 3^2 \cdot 5$ . Each positive integer has a prime factorization, and the fundamental theorem of arithmetic says that it is unique.

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Divisibility definition
Prime number definition
Prime factorization definition

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Tag	`0AEKDFCYFA`
URI	`m2://0AEKDFCYFA`
Aliases	`prime-decomposition`, `integer-factorization`
Kind	definition
Writing	W1 stated
MSC	11A51
Defines	Prime factorization
Uses	Prime number

Prime factorization

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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