Improved exponent bounds for Erdős--Selfridge curves

Given integers $k,\mathrm{\ell }\ge 2$, 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 ${\mathcal{C}}_{k,\mathrm{\ell }}$ with $\mathrm{\ell }$ a prime and $y\mathrm{\ne }0$, then $\mathrm{\ell }$ is bounded (doubly-exponentially) in terms of $k$. We improve this bound on $\mathrm{\ell }$ when $k$ is sufficiently large. Our method uses combinatorial and sieve-theoretic arguments, as well as results on solutions to generalized Fermat equations.