## 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.