On the diophantine equation $An!+Bm!=f(x,y)$

TL;DR

This paper proves finiteness results for solutions to equations involving factorials and polynomial values, extending previous work and providing new unconditional proofs for certain cases.

Contribution

It establishes finiteness of solutions for equations of the form $An!+Bm!$ represented by irreducible homogeneous polynomials, offering new unconditional proofs and exploring reducible cases.

Findings

Finiteness of solutions for $An!+Bm!$ represented by certain polynomials.

Unconditional proofs for specific polynomial degrees.

Analysis of equations involving double factorials.

Abstract

Erd\"os and Obl\'ath proved that the equation $n!\pm m!=x^p$ has only finitely many integer solutions. More general, under the ABC-conjecture, Luca showed that $P(x)=An!+Bm!$ has finitely many integer solutions for polynomials of degree $\geq 3$. For certain polynomials of degree $\geq 2$, this result holds unconditionally. We consider irreducible homogeneous $f(x,y)\in \mathbb{Q}[x,y]$ of degree $\geq 2$ and show that there are only finitely many $n,m$ such that $An!+Bm!$ is represented by $f(x,y)$. As corollaries we get alternative proofs for the unconditional results of Luca. We also discuss the case of certain reducible $f(x,y)$. Furthermore, we study equations of the form $n!!m!!=f(x,y)$ and $n!!m!!=f(x)$.