Polynomial progressions in the generalized twin primes

TL;DR

This paper extends results on primes in polynomial progressions, demonstrating the existence of infinitely many primes in specific polynomial configurations with bounded shifts, using advanced transference techniques.

Contribution

It introduces a novel application of the transference argument to polynomial progressions, generalizing previous results on twin primes and arithmetic progressions.

Findings

Existence of infinitely many primes in polynomial progressions with bounded shifts

Extension of Maynard's theorem to polynomial settings

Application of Tao and Ziegler's transference method to new configurations

Abstract

By Maynard's theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b\leq 246$ such that there are infinitely many primes $p$ such that $p+b$ is also prime. Let $P_1,...,P_t\in \mathbb{Z}[y]$ with $P_1(0)=\cdots=P_t(0)=0$. We use the transference argument of Tao and Ziegler to prove there exist positive integers $x, y,$ and $b \leq 246 $ such that $x+P_1(y),x+P_2(y),...,x+P_t(y)$ and $x+P_1(y)+b,x+P_2(y)+b,...,x+P_t(y)+b$ are all prime. Our work is inspired by Pintz, who proved a similar result for the special case of arithmetic progressions.