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Rational points on Erdős–Selfridge superelliptic curves

  • Michael A. Bennett (a1) and Samir Siksek (a2)

Abstract

Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$
In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell )$ satisfy either $y=0$ or $\ell <\exp (3^{k})$ .

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Rational points on Erdős–Selfridge superelliptic curves

  • Michael A. Bennett (a1) and Samir Siksek (a2)

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