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Rational points on Erdős–Selfridge superelliptic curves
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 11 / November 2016
- Published online by Cambridge University Press:
- 14 July 2016, pp. 2249-2254
- Print publication:
- November 2016
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- Article
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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 In particular, if we assume that
$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$
$\ell$ is prime, then all such triples 
$(x,y,\ell )$ satisfy either 
$y=0$ or 
$\ell <\exp (3^{k})$ .
