Math 2.0

Number theory Theorem

Infinitely many primes congruent to 1 modulo n

Builds on Prime numberCongruence modulo n

Statement

Let nn be a positive integer. Then there are infinitely many prime numbers pp such that p≡1(modn)p \equiv 1 \pmod{n}.

Remarks

This is the case a=1a = 1 of Dirichlet's theorem on arithmetic progressions. The case a=1a = 1 has an elementary proof with cyclotomic polynomials. A prime factor pp of Φn(m)\Phi_n(m) that does not divide nn satisfies p≡1(modn)p \equiv 1 \pmod n. For example, the primes p≡1(mod10)p \equiv 1 \pmod{10} start with 11,31,41,61,7111, 31, 41, 61, 71.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionPrime number, definition, no evidence levelPrime numberdefinitionInfinitely many primes congruent to 1 modulo n, statement, no evidence levelInfinitely many prime…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Divisibility definition
  2. Congruence modulo n definition
  3. Prime number definition
  4. Infinitely many primes congruent to 1 modulo n statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, Examples
About this page
Tag0T1BSD7H3M
URIm2://0T1BSD7H3M
Kindstatement, theorem
WritingW0 stub
MSC11N13
StatusThis copy of the site has no build data. Run m2 status 0T1BSD7H3M.
UsesPrime number, Congruence modulo n

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.