 Dirichlet's theorem on arithmetic progressions

Dirichlet's theorem on arithmetic progressions states that for positive integer $m$ and integer $a$ with $\gcd (a,m)=1$, infinitely many primes satisfy $p\equiv a(\mathrm{mod}m)$.

For example, each of the arithmetic progressions $4n+1$ and $4n+3$ contains infinitely many primes. The theorem establishes infinitude separately in every coprime residue class, rather than only in their union.

The coprimality condition is necessary. If $d=\gcd (a,m)>1$, every integer in the progression is divisible by $d$. A member can then be prime only when it equals $d$ and $d$ itself is prime. This leaves at most one prime value in that residue class. The difficult part of the theorem is sufficiency; its analytic treatment uses Dirichlet L-functions.

Dirichlet's theorem on arithmetic progressions states that if \( a \) and \( d \) are two coprime integers (that is, their greatest common divisor \( \gcd(a, d) = 1 \)), then there are infinitely many prime numbers of the form \( a + nd \), where \( n \) is a non-negative integer.