Products of factorials which are products of factorials

Abstract

In this note, we look at the diophantine equation $$\prod _{i=1}^t{a}_i!=\prod _{j=1}^s{n}_i!,n_1\ge \cdots \ge n_s\ge 2\text{and}{n}_1>a_1\ge a_2\ge \cdots \ge a_t\ge 2.$$ \noindent Let $s<t$. Under the (explicit) abc conjecture, we show that it has only finitely many nontrivial solutions in a certain subset of ${\mathbb{N}}^{t+s}$ of positive asymptotic density.

8 pages, comments are welcome!