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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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The Fermat-type equations $x^5 + y^5 = 2z^p$ or $3z^p$ solved through $\mathbb {Q}$-curves
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by Luis Dieulefait and Nuno Freitas PDF
Math. Comp. 83 (2014), 917-933 Request permission

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
  • MR Author ID: 671876
  • Nuno Freitas
  • Affiliation: Department of Algebra and Geometry, University of Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
  • MR Author ID: 1044711
  • 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. $SFRH/BD/44283/2008$.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 917-933
  • MSC (2010): Primary 11D41
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02731-7
  • MathSciNet review: 3143698