Dirichlet's theorem states that, given an arithmetic progression of terms , for , 2, ..., the series contains an infinite number of primes
if
and
are relatively prime, i.e., . This result had been conjectured by Gauss (Derbyshire
2004, p. 96), but was first proved by Dirichlet (1837).
Dirichlet proved this theorem using Dirichlet L-series, but the proof is challenging enough that, in their classic text on number
theory, the usually explicit Hardy and Wright (1979) report "this theorem
is too difficult for insertion in this book."
, for 
, 2, ..., the series contains an infinite number of primes
if 
and 
are relatively prime, i.e., 
. This result had been conjectured by Gauss (Derbyshire
2004, p. 96), but was first proved by Dirichlet (1837).
