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

TL;DR

This paper proves the existence of infinitely many integers with specific prime factorization properties and small gaps, advancing understanding of prime distributions and related conjectures.

Contribution

The authors establish new results on the distribution of integers with prescribed prime divisor counts, extending previous work and confirming conjectures for almost primes.

Findings

Infinitely many integers with  prime divisors in both x and x+1

Results hold for arbitrary shifts b, not just 1

Sharpens earlier results by Heath-Brown, Pinner, and Schlage-Puchta

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 $E_2$-values; i.e., values that are products of exactly two primes. We use that result to prove that there are inifinitely many integers $x$ that simultaneously satisfy $$\omega(x)=\omega(x+1)=4, \Omega(x)=\Omega(x+1)=5, \text{and} d(x)=d(x+1)=24.$$ Here, $\omega(x), \Omega(x), d(x)$ represent the number of prime divisors of $x$, the number of prime power divisors of $x$, and the number of divisors of $x$, respectively. We also prove similar theorems where $x+1$ is replaced by $x+b$ for an arbitrary positive integer $b$. Our results sharpen earlier work of Heath-Brown, Pinner, and Schlage-Puchta.