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## Acta Arithmetica

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## Small gaps between three almost primes and almost prime powers

### Volume 203 / 2022

Acta Arithmetica 203 (2022), 1-18 MSC: Primary 11N36; Secondary 11N25. DOI: 10.4064/aa210325-27-11 Published online: 24 March 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 $x$.

#### Authors

• Daniel A. GoldstonDepartment of Mathematics and Statistics
San José State University
1 Washington Sq.
San Jose, CA 95192-0103, USA
e-mail
• Apoorva PanidapuDepartment of Mathematics and Statistics
San José State University
1 Washington Sq.
San Jose, CA 95192-0103, USA
e-mail
• Jordan SchettlerDepartment of Mathematics and Statistics
San José State University
1 Washington Sq.
San Jose, CA 95192-0103, USA
e-mail

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