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

TL;DR

This paper investigates the existence and finiteness of integer solutions for equations involving sums of products of consecutive integers, extending classical results and exploring additive variants with specific power exponents.

Contribution

It generalizes the Erdős-Selfridge equation to sums of products and establishes finiteness results, also proving infinite solutions for certain additive equations with specific exponents.

Findings

Finiteness of solutions for certain product sum equations when n ≥ 2 and a₁ ≥ 2.

Existence of infinitely many solutions for specific additive equations with m=2 or 3.

Extension of classical Diophantine equations to new sum-of-products forms.

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.