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Rational points on Erdős–Selfridge superelliptic curves
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Bennett, Michael A. and Siksek, Samir (2016) Rational points on Erdős–Selfridge superelliptic curves. Compositio Mathematica, 152 (11). pp. 2249-2254. doi:10.1112/S0010437X16007569 ISSN 0010-437X.
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Official URL: https://doi.org/10.1112/S0010437X16007569
Abstract
Given k⩾2k⩾2, we show that there are at most finitely many rational numbers xx and y≠0y≠0 and integers ℓ⩾2ℓ⩾2 (with (k,ℓ)≠(2,2)(k,ℓ)≠(2,2)) for which $$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$ In particular, if we assume that ℓℓ is prime, then all such triples (x,y,ℓ)(x,y,ℓ) satisfy either y=0y=0 or ℓ<exp(3k)ℓ<exp(3k).
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | Compositio Mathematica | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 0010-437X | ||||||||
Official Date: | November 2016 | ||||||||
Dates: |
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Volume: | 152 | ||||||||
Number: | 11 | ||||||||
Page Range: | pp. 2249-2254 | ||||||||
DOI: | 10.1112/S0010437X16007569 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 14 March 2016 | ||||||||
Date of first compliant Open Access: | 8 February 2017 | ||||||||
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