The primes contain arbitrarily long polynomial progressions

Abstract

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P₁, …, P_k ∈ Z[m] in one unknown m with P₁(0) = … = P_k(0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with $1 \leqslant m \leqslant x^\varepsilon$, such that x + P₁(m), …, x + P_k(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P_j = (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.