Power values of sums of certain products of consecutive integers and related results

Abstract

Let $n$ be a non-negative integer and put $p_{n}(x)=\prod_{i=0}^{n}(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$ y^m=p_{n}(x)+\sum_{i=1}^{k}p_{a_{i}}(x), $$ where $m\in\N_{\geq 2}$ and $a_{1}<a_{2}<\ldots <a_{k}<n$. This equation can be considered as a generalization of the Erd\H{o}s-Selfridge Diophantine equation $y^m=p_{n}(x)$. We present some general finiteness results concerning the integer solutions of the above equation. In particular, if $n\geq 2$ with $a_{1}\geq 2$, then our equation has only finitely many solutions in integers. In the second part of the paper we study the equation $$ y^m=\sum_{i=1}^{k}p_{a_{i}}(x_{i}), $$ for $m=2, 3$, which can be seen as an additive version of the equation considered by Erd\H{o}s and Graham. In particular, we prove that if $m=2, a_{1}=1$ or $m=3, a_{2}=2$, then for each $k-1$ tuple of positive integers $(a_{2},\ldots, a_{k})$ there are infinitely many solutions in integers.