Intersective polynomials and the primes

TL;DR

This paper demonstrates that for any intersective polynomial, the difference between two primes in a dense subset of primes can be expressed as that polynomial evaluated at some integer, extending results to Chen primes.

Contribution

It leverages recent results to show that intersective polynomials can describe prime differences within dense prime subsets, including Chen primes.

Findings

Prime differences in dense subsets can be represented by intersective polynomials.

The results extend to Chen primes, broadening the scope of prime difference patterns.

Uses advanced results of Green-Tao and Lucier to establish these properties.

Abstract

Intersective polynomials are polynomials in $\Z[x]$ having roots every modulus. For example, $P_1(n)=n^2$ and $P_2(n)=n^2-1$ are intersective polynomials, but $P_3(n)=n^2+1$ is not. The purpose of this note is to deduce, using results of Green-Tao \cite{gt-chen} and Lucier \cite{lucier}, that for any intersective polynomial $h$, inside any subset of positive relative density of the primes, we can find distinct primes $p_1, p_2$ such that $p_1-p_2=h(n)$ for some integer $n$. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number $p$ such that $p+2$ is the product of at most 2 primes).