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Mordell’s equation: a classical approach

Part of: Arithmetic algebraic geometry Diophantine equations

Published online by Cambridge University Press:  01 September 2015

Michael A. Bennett  and
Amir Ghadermarzi
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2 email bennett@math.ubc.ca
Amir Ghadermarzi
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2 email amir@math.ubc.ca

Abstract

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We solve the Diophantine equation $Y^{2}=X^{3}+k$ for all nonzero integers $k$ with $|k|\leqslant 10^{7}$ . Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

MSC classification

Primary: 11D25: Cubic and quartic equations 11G05: Elliptic curves over global fields

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 633 - 646
Copyright
© The Author(s) 2015 

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