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arXiv · 1510.05376

Rational Points on Erdos-Selfridge Superelliptic Curves

Michael Bennett·Samir Siksek
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Abstract

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

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BibTeXRIS

Michael Bennett, Samir Siksek. 2015-10-19. Rational Points on Erdos-Selfridge Superelliptic Curves. https://doi.org/10.1112/s0010437x16007569

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