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Rational points on Erdős–Selfridge superelliptic curves

Part of: Discontinuous groups and automorphic forms Diophantine equations

Published online by Cambridge University Press:  14 July 2016

Michael A. Bennett  and
Samir Siksek
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 email [email protected]
Samir Siksek
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email [email protected]
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Abstract

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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

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

Keywords

superelliptic curvesGalois representationsFrey curvemodularitylevel lowering

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11D41: Higher degree equations; Fermat's equation 11F80: Galois representations 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 11 , November 2016 , pp. 2249 - 2254
Copyright
© The Authors 2016 

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