Polynomial Differences in the Primes

TL;DR

This paper uses the Hardy-Littlewood Circle Method to derive an asymptotic formula for counting prime pairs with differences in polynomial images, extending to linear combinations of primes.

Contribution

It introduces a novel application of the Hardy-Littlewood Circle Method to polynomial differences among primes and generalizes to linear combinations.

Findings

Derived an asymptotic formula for prime pairs with polynomial differences

Extended results to differences expressed as linear combinations of primes

Provides a new approach to understanding prime gaps and differences

Abstract

We establish, utilizing the Hardy-Littlewood Circle Method, an asymptotic formula for the number of pairs of primes whose differences lie in the image of a fixed polynomial. We also include a generalization of this result where differences are replaced with any integer linear combination of two primes.