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The Fermat-type equations $ x^5 + y^5 = 2z^p$ or $ 3z^p$ solved through $ \mathbb{Q}$-curves


Authors: Luis Dieulefait and Nuno Freitas
Journal: Math. Comp. 83 (2014), 917-933
MSC (2010): Primary 11D41
DOI: https://doi.org/10.1090/S0025-5718-2013-02731-7
Published electronically: June 10, 2013
MathSciNet review: 3143698
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Abstract: We solve the Diophantine equations $ x^5 + y^5 = dz^p$ with $ d=2, 3$ for a set of prime numbers of density $ 3/4$. The method consists of relating a possible solution to another Diophantine equation and solving the latter via a generalized modular technique. Indeed, we will apply a multi-Frey technique with two $ \mathbb{Q}$-curves along with a new technique for eliminating newforms.


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Additional Information

Luis Dieulefait
Affiliation: Department of Algebra and Geometry, University of Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain

Nuno Freitas
Affiliation: Department of Algebra and Geometry, University of Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain

DOI: https://doi.org/10.1090/S0025-5718-2013-02731-7
Received by editor(s): May 24, 2011
Received by editor(s) in revised form: November 26, 2011, December 15, 2011, January 13, 2012, March 6, 2012, and June 2, 2012
Published electronically: June 10, 2013
Additional Notes: The first author’s research was supported by project MICINN MTM2009-07024 from MECD, Spain; and ICREA Academia Research Prize.
The second author’s research was supported by a scholarship from Fundaçao para a Ciência e a Tecnologia, Portugal, reference no. $SFRH/BD/44283/2008$.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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