Billerey, Nicolas, Chen, Imin, Dieulefait, Luis and Freitas, Nuno (2019) A multi-Frey approach to Fermat equations of signature (r,r,p). Transactions of the American Mathematical Society, 371 (12). pp. 8651-8677. doi:10.1090/tran/7477 ISSN 0002-9947.

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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$.

Item Type:	Journal Article
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Transactions of the American Mathematical Society
Publisher:	American Mathematical Society
ISSN:	0002-9947
Official Date:	7 March 2019
Dates:	Date Event 7 March 2019 Published 30 November 2017 Accepted
Volume:	371
Number:	12
Page Range:	pp. 8651-8677
DOI:	10.1090/tran/7477
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Copyright Holders:	© Copyright 2019 American Mathematical Society

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