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Mathematics of Computation

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Solving Fermat-type equations $ x^5+y^5=dz^p$

Authors: Nicolas Billerey and Luis V. Dieulefait
Journal: Math. Comp. 79 (2010), 535-544
MSC (2000): Primary 11F11, 11D41, 14H52; Secondary 11D59
Published electronically: July 22, 2009
MathSciNet review: 2552239
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Abstract: In this paper, we are interested in solving the Fermat-type equations $ x^5+y^5=dz^p$, where $ d$ is a positive integer and $ p$ a prime number $ \ge 7$. We describe a new method based on modularity theorems which allows us to improve all earlier results for this equation. We finally discuss the present limits of the method by looking at the case $ d=3$.

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Additional Information

Nicolas Billerey
Affiliation: Université Pierre et Marie Curie – Paris 6, UMR 7586, Case 247, 4, place Jussieu, Institut de Mathématiques, 75252 Paris, France

Luis V. Dieulefait
Affiliation: Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, (08007) Barcelona, Spain

Keywords: Modular forms, Fermat's equation, elliptic curves, Thue-Mahler equations
Received by editor(s): July 10, 2008
Received by editor(s) in revised form: January 28, 2009
Published electronically: July 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.