Green–Tao theorem

In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi's theorem. The conjecture was probably stated at least in 1770.

Let ${\displaystyle \pi (N)}$ denote the number of primes less than or equal to ${\displaystyle N}$. If ${\displaystyle A}$ is a subset of the prime numbers such that

then for all positive integers ${\displaystyle k}$, the set ${\displaystyle A}$ contains infinitely many arithmetic progressions of length ${\displaystyle k}$. In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions.

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