---
tag: 0T1BSD7H3M
kind: statement
role: theorem
name: Infinitely many primes congruent to 1 modulo n
msc: [11N13]
uses: [0SN0MAEKYX, 0TPW25CBN6]
sources:
  - url: https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions
    locator: Examples
---

## Statement

Let $n$ be a positive integer.
Then there are infinitely many [[prime numbers|0SN0MAEKYX]] $p$ such that $p \equiv 1 \pmod{n}$.

## Remarks

This is the case $a = 1$ of [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].
The case $a = 1$ has an elementary proof with cyclotomic polynomials. A prime factor $p$ of $\Phi_n(m)$ that does not divide $n$ satisfies $p \equiv 1 \pmod n$.
For example, the primes $p \equiv 1 \pmod{10}$ start with $11, 31, 41, 61, 71$.

## Explanations

### L1 human

Fix a number $n$, for example $n = 6$. Then infinitely many primes leave the remainder $1$ when you divide them by $6$.
The first ones are $7, 13, 19, 31, 37$.