Bounded length intervals containing two primes and an almost-prime II

TL;DR

This paper proves the existence of infinitely many bounded intervals that contain two primes and an almost-prime with at most 31 prime factors, advancing understanding of prime distributions within short intervals.

Contribution

It demonstrates that infinitely many intervals of length at most 10^8 contain two primes and an almost-prime, extending previous results on prime gaps and almost-primes.

Findings

Infinitely many intervals of length ≤ 10^8 contain two primes and an almost-prime.

The result holds unconditionally, without assuming conjectures like Elliott-Halberstam.

Provides new bounds on the distribution of primes and almost-primes in short intervals.

Abstract

Zhang has shown there are infinitely many intervals of bounded length containing two primes. It appears that the current techniques cannot prove that there are infinitely many intervals of bounded length containing three primes, even if strong conjectures such as the Elliott-Halberstam conjecture are assumed. We show that there are infinitely many intervals of length at most $10^8$ which contain two primes and a number with at most 31 prime factors.