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.
Articles by others on the same topic
Dirichlet's theorem on arithmetic progressions states that for positive integer and integer with , infinitely many primes satisfy .
For example, each of the arithmetic progressions and 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 , every integer in the progression is divisible by . A member can then be prime only when it equals and 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.