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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 satisfyWe prove several other similar theorems. Our results sharpen earlier works by Heath-Brown and Schlage-Puchta.

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