Small Gaps Between Three Almost Primes and Almost Prime Powers

TL;DR

This paper investigates small gaps between almost primes and almost prime powers, establishing new bounds and conditional results under the Elliott-Halberstam conjecture, with implications for the distribution of divisors in integers.

Contribution

It proves the existence of infinitely many small gaps between specific types of almost primes and almost prime powers, improving bounds under certain conjectures.

Findings

Infinitely many triples of $E_2$-numbers within gap 32.

Infinitely many triples of $E_3$-numbers within gap 15.

Conditional improvements to gaps under Elliott-Halberstam conjecture.

Abstract

A positive integer is called an $E_j$-number if it is the product of $j$ distinct primes. We prove that there are infinitely many triples of $E_2$-numbers within a gap size of $32$ and infinitely many triples of $E_3$-numbers within a gap size of $15$. Assuming the Elliot-Halberstam conjecture for primes and $E_2$-numbers, we can improve these gaps to $12$ and $5$, respectively. We can obtain even smaller gaps for almost primes, almost prime powers, or integers having the same exponent pattern in the their prime factorizations. In particular, if $d(x)$ denotes the number of divisors of $x$, we prove that there are integers $a,b$ with $1\leq a < b \leq 9$ such that $d(x)=d(x+a)=d(x+b) = 192$ for infinitely many $x$. Assuming Elliot-Halberstam, we prove that there are integers $a,b$ with $1\leq a < b \leq 4$ such that $d(x)=d(x+a)=d(x+b) = 24$ for infinitely many $x$.