Dirichlet's theorem on arithmetic progressions states that if two numbers, a and d, are coprime (they have no common factors other than 1), then the arithmetic progression a, a+d, a+2d, a+3d, \ldots contains infinitely many prime numbers. This means that you can find prime numbers in sequences formed by adding a constant d to a starting number a. The theorem was proven by the mathematician Peter Gustav Lejeune Dirichlet in the 19th century. It is significant in number theory as it extends the understanding of prime distribution beyond simple sequences, showing that primes can be found in structured patterns, not just randomly among the integers.