Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdos on Consecutive Integers

Abstract

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E 2 -values; that is, values that are products of exactly two primes. We use this result to prove that there are infinitely many positive integers x such that both x and x + 1 have prime factorizations of the form p 1 2 p 2 p 3 p 4 . Consequently, there are infinitely many integers x that simultaneously satisfy We prove several other similar theorems. Our results sharpen earlier works by Heath-Brown and Schlage-Puchta.