Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers II

Abstract

We show that for any positive integer n, there is some fixed A such that d(x)=d(x+n)=A infinitely often where d(x) denotes the number of divisors of x. In fact, we establish the stronger result that both x and x+n have the same fixed exponent pattern for infinitely many x. Here the exponent pattern of an integer x>1 is the multiset of nonzero exponents which appear in the prime factorization of x.