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Rational points on Erdős–Selfridge superelliptic curves
Published online by Cambridge University Press: 14 July 2016
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
$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$
$\ell$ is prime, then all such triples
$(x,y,\ell )$ satisfy either
$y=0$ or
$\ell <\exp (3^{k})$.
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- © The Authors 2016
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