Bounded length intervals containing two primes and an almost-prime

TL;DR

This paper extends results on small intervals containing primes by showing that, under certain distribution hypotheses, such intervals also contain almost-primes with bounded prime factors, with explicit bounds for the interval length.

Contribution

It introduces new bounds for intervals containing two primes and an almost-prime with a limited number of prime factors, under the assumption of prime distribution levels.

Findings

Existence of infinitely many intervals with two primes and an almost-prime under distribution hypotheses.

Explicit bounds for interval length and prime factors of almost-primes.

Conditions under which small intervals contain specific prime and almost-prime configurations.

Abstract

Goldston, Pintz and Y\i ld\i r\i m have shown that if the primes have `level of distribution' $\theta$ for some $\theta>1/2$ then there exists a constant $C(\theta)$, such that there are infinitely many integers $n$ for which the interval $[n,n+C(\theta)]$ contains two primes. We show under the same assumption that for any integer $k\ge 1$ there exists constants $D(\theta,k)$ and $r(\theta,k)$, such that there are infinitely many integers $n$ for which the interval $[n,n+D(\theta,k)]$ contains two primes and $k$ almost-primes, with all of the almost-primes having at most $r(\theta,k)$ prime factors. If $\theta$ can be taken as large as $1-\epsilon$, and provided that numbers with 2, 3, or 4 prime factors also have level of distribution $1-\epsilon$, we show that there are infinitely many integers $n$ such that the interval $[n,n+90]$ contains 2 primes and a number with at most 4 prime factors.