Explicit Bounds For The Diophantine Equation A!B! = C!

TL;DR

This paper investigates the Diophantine equation A!B! = C!, providing new explicit bounds and computational evidence supporting the conjecture that the only nontrivial solution is (6, 7, 10).

Contribution

It offers improved explicit bounds on the parameters and extends the computational verification of the conjecture up to B ≤ 10,3000.

Findings

Confirmed the conjecture for B ≤ 10,3000.

Derived tighter bounds on C - B involving log log B.

Provided explicit bounds improving previous estimates.

Abstract

A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A $\le$ B $\le$ C -- 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6. Several estimates on the relative size of the parameters are known, such as the one given by Erd{\"o}s C -- B $\le$ 5 log log C, or the one given by Bhat and Ramachandra C --B $\le$ (1/ log 2+o(1)) log log C. We check the conjecture for B $\le$ 10 3000 and give better explicit bounds such as C -- B $\le$ log log(B+1) log 2 -- 0.8803.