Improved exponent bounds for Erdős--Selfridge curves

Abstract

Given integers $k,\ell \ge 2$, define the Erdős--Selfridge curve \begin{align*} \mathcal{C}_{k,\ell} : y^\ell = x(x+1)\cdots (x+k-1). \end{align*} Bennett and Siksek proved that if $(x,y)$ is a rational point on ${\mathcal{C}}_{k,\ell }$ with $\ell$ a prime and $y\ne 0$, then $\ell$ is bounded (doubly-exponentially) in terms of $k$. We improve this bound on $\ell$ when $k$ is sufficiently large. Our method uses combinatorial and sieve-theoretic arguments, as well as results on solutions to generalized Fermat equations.

19 pages