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Limits RegistryBen Green and Terence Tao proved in a 2004 paper (published in Annals of Mathematics in 2008) that the prime numbers contain arithmetic progressions of every finite length, despite primes becoming increasingly sparse. The proof combines ergodic theory and additive combinatorics, extending Szemerédi's theorem on arithmetic progressions in dense sets to the 'sparse' set of primes via a transference principle.
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Ben Green and Terence Tao proved in a 2004 paper (published in Annals of Mathematics in 2008) that the prime numbers contain arithmetic progressions of every finite length, despite primes becoming increasingly sparse. The proof combines ergodic theory and additive combinatorics, extending Szemerédi's theorem on arithmetic progressions in dense sets to the 'sparse' set of primes via a transference principle.
For any positive integer k, there exist k prime numbers in arithmetic progression (i.e., k primes p, p+d, p+2d, ..., p+(k-1)d for some common difference d).
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Limits Registry. LR-GREEN-TAO-THEOREM. Green-Tao Theorem on primes in arithmetic progression. 2026.