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

Erd\"os 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 \(\geq 3\). 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 \(n!!m!!=f(x)\).