Bounded length intervals containing two primes and an almost-prime

Abstract

Goldston, Pintz and Yıldırım 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.

13 Pages

References in corpus (1)

• Small gaps between products of two primes

Cited by in corpus (4)

• The "bounded gaps between primes" Polymath project - a retrospective
• Bounded gaps between primes of the special form
• Variants of the Selberg sieve, and bounded intervals containing many primes
• Revisiting Generalized Bertand's Postulate and Prime Gaps