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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A multi-Frey approach to Fermat equations of signature $ (r,r,p)$

Authors: Nicolas Billerey, Imin Chen, Luis Dieulefait and Nuno Freitas
Journal: Trans. Amer. Math. Soc. 371 (2019), 8651-8677
MSC (2010): Primary 11D41; Secondary 11F80, 11G05
Published electronically: March 7, 2019
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Abstract: In this paper, we give a resolution of the generalized Fermat equations

$\displaystyle x^5 + y^5 = 3 z^n$$\displaystyle \quad \text {and}\quad x^{13} + y^{13} = 3 z^n,$    

for all integers $ n \ge 2$ and all integers $ n \ge 2$ which are not a power of $ 7$, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $ n$.

We also give a number of results for the equations $ x^5 + y^5 = d z^n$, where $ d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem and which uses a new application of level raising at $ p$ modulo $ p$.

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Additional Information

Nicolas Billerey
Affiliation: Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Imin Chen
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

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

Nuno Freitas
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2 Canada

Keywords: Fermat equations, modular method, multi-Frey
Received by editor(s): October 8, 2017
Received by editor(s) in revised form: November 30, 2017
Published electronically: March 7, 2019
Additional Notes: The first author acknowledges the financial support of ANR-14-CE-25-0015 Gardio.
The fourth author was partly supported by the grant Proyecto RSME-FBBVA $2015$ José Luis Rubio de Francia.
Article copyright: © Copyright 2019 American Mathematical Society