We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P1, …, Pk ∈ Z[m] in one unknown m with P1(0) = … = Pk(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 + Pk(m) are simultaneously prime. The arguments are based on those in , which treated the linear case Pj = (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.
The second author was partially supported by NSF grant DMS-0111298. This work was initiated at a workshop held at the CRM in Montreal. The authors would like to thank the CRM for their hospitality.
"The primes contain arbitrarily long polynomial progressions." Acta Math. 201 (2) 213 - 305, 2008. https://doi.org/10.1007/s11511-008-0032-5