Small gaps between almost primes, the parity problem, and some conjectures of Erd\H{o}s on consecutive integers II

Abstract

This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erd\H{o}s-Mirksy conjecture which states that there are infinitely many positive integers \(x\) such that \(d(x)=d(x+1)\) where \(d(x)\) denotes the number of divisors of \(x\). This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values \(d(x)\) for such \(x\). In particular, one could not conclude that there was any particular value \(A\) for which \(d(x)=d(x+1)=A\) infinitely often. In the previous paper arXiv:0803.2636, the authors showed that there are infinitely many positive integers \(x\) such that both \(x\) and \(x+1\) have exponent pattern \(\{2,1,1,1\}\), so \(d(x)=d(x+1)=24\). Similar results were known for certain shifts \(n\), i.e., \(x\) and \(x+n\) have the same fixed exponent pattern infinitely often. This was done for shifts \(n\) which are either even or not divisible by the product of a pair of twin primes. The goal of this paper is to give simple proofs of results on exponent patterns for an arbitrary shift \(n\).