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