Small gaps between three almost primes and almost prime powers
Volume 203 / 2022
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 Elliott–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 $1\leq a \lt b \leq 9$ such that $d(x)=d(x+a)=d(x+b) = 192$ for infinitely many $x$. Assuming Elliott–Halberstam, we prove that there are integers $a,b$ with $1\leq a \lt b\leq 4$ such that $d(x)=d(x+a)=d(x+b)=24$ for infinitely many $x$.
