On small gaps between primes and almost prime powers

TL;DR

This paper proves that the union of primes and certain almost prime powers contains infinitely many bounded gaps, extending previous results by including a thinner set of numbers with nearly equal prime factors.

Contribution

It demonstrates that adding specific almost prime powers to primes guarantees infinitely many bounded gaps in the combined set.

Findings

Union of primes and almost prime squares with two nearly equal prime factors has bounded gaps.

Union of primes and almost prime cubes with three nearly equal prime factors has bounded gaps.

Extends previous work on prime gaps by including almost prime powers.

Abstract

In a recent joint work with D.A. Goldston and C.Y. Yildirim we just missed by a hairbreadth a proof that bounded gaps between primes occur infinitely often. In the present work it is shown that adding to the primes a much thinner set, called almost prime powers, the union of the set of primes and almost prime powers contains already infinitely many bounded gaps. More precisely it is shown that if we add to the set of primes either almost prime squares having exactly two, nearly equal prime factors or if we add to the set of primes almost prime cubes having exactly three, nearly equal prime factors, then the resulting set contains already infinitely many bounded gaps.