Daniel A. Goldston, Sidney W. Graham, Janos Pintz, Cem Y. Yıldırım; Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdős on Consecutive Integers. Int Math Res Notices 2011; 2011 (7): 1439-1450. doi: 10.1093/imrn/rnq124
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 E2-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 p12p2p3p4. 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.