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

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