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On the diophantine equation $An!+Bm!=f(x,y)$
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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 $n!!m!!=f(x)$.
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A comment on the number of $k$-th powers inside arithmetic progressions
For an arithmetic progression of length N whose step q is at most N^r, the number of k-th powers among its terms is O(N^{1/k+ε}) for large N.
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