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On the equations a2−2b6=cp and a2−2=cp

Part of: Diophantine equations Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 May 2012

Imin Chen
Imin Chen*
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6 Canada (email: ichen@math.sfu.ca)

Abstract

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We study the equation a2−2b6=cp and its specialization a2−2=cp, where p is a prime, using the modular method. In particular, we use a ℚ-curve defined over for which the solution (a,b,c)=(±1,±1,−1) gives rise to a CM-form. This allows us to apply the modular method to resolve the equation a2−2b6=cp for p in certain congruence classes. For the specialization a2−2=cp, we use the multi-Frey technique of Siksek to obtain further refined results.

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation 14G05: Rational points 11G05: Elliptic curves over global fields 11D61: Exponential equations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 158 - 171
Copyright
Copyright © London Mathematical Society 2012

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