A step towards proving de Polignac's Conjecture

TL;DR

This paper presents a conditional proof that, for certain even gaps, there are infinitely many prime pairs with that gap, advancing the understanding of de Polignac's conjecture within a specific range.

Contribution

It introduces a novel approach linking co-prime number sets to prime gaps, providing a partial proof of de Polignac's conjecture for specific even gaps.

Findings

Proves the existence of infinitely many prime pairs for certain even gaps under specific conditions.

Establishes a connection between co-prime sets and prime gap distribution.

Provides a framework that could lead to a full proof of de Polignac's conjecture with further work.

Abstract

Consider the set of all natural numbers that are co-prime to primes less than or equal to a given prime. Then given a consecutive pair of numbers in that set with an arbitrary even gap, we prove there exists an unbounded number of actual prime pairs with that same gap. This conditional proof of de Polignac's conjecture constitutes a proof for a range of known gaps, but the full conjecture requires additional proof that such number pairs exist for all even gaps.