Rational Points on Erdos-Selfridge Superelliptic Curves

TL;DR

This paper proves finiteness results for rational solutions to certain superelliptic equations involving products of consecutive integers, with implications for the structure of rational points on these curves.

Contribution

It establishes finiteness of rational solutions for a class of superelliptic equations, extending understanding of rational points on these algebraic curves.

Findings

Finitely many rational solutions for the equations considered.

If ll is prime, solutions satisfy specific bounds.

The results exclude infinite families of solutions for the given equations.

Abstract

Given $k \geq 2$, we show that there are at most finitely many rational numbers $x$ and $y \neq 0$ and integers $\ell \geq 2$ (with $(k,\ell) \neq (2,2)$) for which $$ x (x+1) \cdots (x+k-1) = y^\ell. $$ In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell)$ satisfy either $y=0$ or $\log \ell < 3^k$.