The problem
Theorem: the set of prime numbers contains arithmetic progressions of length k for every positive integer k. Infinitely many such progressions exist for each k.
The Green–Tao Theorem
historicReferences
History & significance
Euler observed in 1770 that primes cluster in patterns. Lagrange, Waring and others studied special cases. The modern framework required two ingredients Szemerédi's 1975 theorem (dense sets contain arbitrary long APs) and a way to transfer that result from dense sets to the sparse primes. Tao and Green built the transference ('relative Szemerédi') machinery in their 2004 Annals paper, using pseudorandom majorants and Gowers uniformity norms.
The resolution (human proof)
Proven by Ben Green and Terence Tao, 2004 (Annals of Mathematics). The proof decomposes the primes into a dense part (via W-trick) and a pseudorandom error term controlled by Gowers norms; Szemerédi then applies to the dense component. The transference principle they invented launched a subfield — subsequent work includes Maynard's bounded-gap clusters and the Erdős-distinct-primes extensions. One of the youngest theorems on our historic shelf, and already foundational.