Products of factorials which are products of factorials

TL;DR

This paper investigates a specific Diophantine equation involving products of factorials, demonstrating under the abc conjecture that only finitely many solutions exist in a particular subset of natural numbers.

Contribution

It establishes finiteness of solutions for a factorial product equation under the abc conjecture, extending understanding of factorial Diophantine equations.

Findings

Finiteness of solutions under the abc conjecture

Explicit conditions for solution finiteness

Solutions form a set of positive asymptotic density

Abstract

In this note, we look at the diophantine equation $$ \prod_{i=1}^ta_i!=\prod_{j=1}^sn_i!, \quad n_1\geq \cdots \geq n_s\geq 2 \quad \textnormal{and}\quad n_1>a_1\geq a_2\geq\cdots \geq a_t\geq2. $$ \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.