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VARIANTS OF ERDŐS–SELFRIDGE SUPERELLIPTIC CURVES AND THEIR RATIONAL POINTS

Part of: Diophantine equations

Published online by Cambridge University Press:  03 April 2018

Pranabesh Das ,
Shanta Laishram  and
N. Saradha
Pranabesh Das
Affiliation:
Stat-Math Unit, Indian Statistical Institute Delhi Centre, New Delhi 110016, India email [email protected]
Shanta Laishram
Affiliation:
Stat-Math Unit, Indian Statistical Institute Delhi Centre, New Delhi 110016, India email [email protected]
N. Saradha
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India email [email protected]
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Abstract

For the superelliptic curves of the form

$$\begin{eqnarray}(x+1)\cdots (x+i-1)(x+i+1)\cdots (x+k)=y^{\ell }\end{eqnarray}$$
with $y\neq 0$, $k\geqslant 3$, $\ell \geqslant 2,$ a prime and for $i\in [2,k]\setminus \unicode[STIX]{x1D6FA}$, we show that $\ell <\text{e}^{3^{k}}.$ Here $\unicode[STIX]{x1D6FA}$ denotes the interval $[p_{\unicode[STIX]{x1D703}},(k-p_{\unicode[STIX]{x1D703}}))$, where $p_{\unicode[STIX]{x1D703}}$ is the least prime greater than or equal to $k/2$. Bennett and Siksek obtained a similar bound for $i=1$ in a recent paper.

MSC classification

Primary: 11D61: Exponential equations
Type
Research Article
Information
Mathematika , Volume 64 , Issue 2 , 2018 , pp. 380 - 386
Copyright
Copyright © University College London 2018 

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