Products of factorials which are powers
Volume 190 / 2019
Acta Arithmetica 190 (2019), 339-350
MSC: 11D41, 11D85.
DOI: 10.4064/aa171008-16-10
Published online: 29 July 2019
Abstract
Extending earlier research of Erdős and Graham, we consider the problem of products of factorials yielding perfect powers. On the one hand, we describe how the representability of $\ell$th powers behaves when the number of factorials is smaller than, equal to or larger than $\ell$, respectively. On the other hand, we investigate for which fixed $n=b_1$ it is possible to find integers $b_2,\dots,b_k$ at most $b_1$ (obeying certain conditions) such that $b_1!\cdots b_k!$ is a perfect power. Here we distinguish the cases where the factorials may be repeated or are distinct.
