Bennett-Siksek conjecture on rational points on Erdős-Selfridge curves

Let $k,\ell\geq 2$ be integers, and consider the Erdős-Selfridge curve

$$y^\ell=x(x+1)\cdots(x+k-1).$$

The trivial rational points are those with $y=0$; the claim classifies all rational points $(x,y)\in\mathbb{Q}^2$.

Bennett-Siksek conjecture. The only rational points satisfy $y=0$, or are one of the following:

• $(x,y,k,\ell)=\left(\frac{a^2}{b^2-a^2},\frac{ab}{b^2-a^2},2,2\right)$ for integers $a\neq\pm b$;
• $(x,y,k,\ell)=\left(\frac{1-2j}{2},\pm 2^{-j}\prod_{i=1}^j(2i-1),2j,2\right)$ for an even integer $j\geq 2$;
• $(x,y,k,\ell)=\left(-\frac43,\frac23,3,3\right)$; or
• $(x,y,k,\ell)=\left(-\frac23,-\frac23,3,3\right)$.

For $k+\ell\geq 7$, the associated smooth projective curve has genus at least $2$, so Faltings's theorem gives finiteness of its rational points, but not this explicit classification. The statement is presented in the source as the corrected version of Sander's conjecture and remains unresolved there.