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

Abstract

Erdös and Obláth 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 $\ge 3$. For certain polynomials of degree $\ge 2$, this result holds unconditionally. We consider irreducible homogeneous $f(x,y)\in \mathbb{Q}[x,y]$ of degree $\ge 2$ 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 $n!!m!!=f(x)$.

12 pages, comments are welcome! arXiv admin note: substantial text overlap with arXiv:2308.11002