= Dirichlet's theorem on arithmetic progressions
{c}
{wiki}
= Dirichlet theorem on primes in arithmetic progressions
{c}
{synonym}
<Dirichlet's theorem on arithmetic progressions> states that for positive <integer> $m$ and <integer> $a$ with $\gcd(a,m)=1$, infinitely many <primes> satisfy $p\equiv a\pmod m$.
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>.
Back to article page
