Rational Points on Erdos-Selfridge Superelliptic Curves

Abstract

Given $k\ge 2$, we show that there are at most finitely many rational numbers $x$ and $y\ne 0$ and integers $\ell \ge 2$ (with $(k,\ell )\ne (2,2)$) 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 $\log \ell <3^k$.