Dirichlet's theorem on arithmetic progressions

Every arithmetic progression of positive integers with first term coprime to its common difference contains infinitely many primes. The theorem also gives their asymptotic distribution through arithmetic progressions.

Dirichlet character: A periodic, completely multiplicative function that encodes residue classes modulo an integer. Characters modulo the common difference isolate primes in the progression’s residue class.
Arithmetic progression: A sequence whose consecutive terms differ by a fixed constant. The theorem concerns prime terms in this basic sequence.
Peter Gustav Lejeune Dirichlet: A German mathematician whose work established foundational results in number theory and analysis. He proved the theorem in 1837 using characters and analytic arguments.
Primality testing: The problem of determining whether a given integer is prime. The theorem guarantees an unlimited supply of candidate primes in eligible progressions, not a test for any one candidate.
Euclid's theorem: The theorem that there are infinitely many prime numbers. It gives unrestricted infinitude, while Dirichlet specifies which residue classes contain infinitely many primes.
Dirichlet L-function: A complex function formed by weighting the Dirichlet series of the Riemann zeta function with a Dirichlet character. Nonvanishing of these functions at 1 underpins the theorem’s infinitude claim.
Coprime integers: Two integers whose greatest common divisor is 1. Coprimality of the first term and difference is precisely the eligibility condition.
Leonhard Euler: An eighteenth-century mathematician who made foundational contributions to analysis and number theory. Euler’s work on prime-producing series helped shape the problem Dirichlet solved.
Sophie Germain prime: A prime p for which 2p + 1 is also prime. Such primes impose simultaneous primality constraints beyond a single arithmetic progression.
Twin prime conjecture: The conjecture that infinitely many pairs of primes differ by 2. Dirichlet handles one fixed residue class, not the simultaneous primality constraints in prime pairs.

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