Rational Points on Erdos-Selfridge Superelliptic Curves

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 $\log \ell < 3^k$.