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

Billerey, Nicolas; Chen, Imin; Dieulefait, Luis; Freitas, Nuno

doi:10.1090/tran/7477

A multi-Frey approach to Fermat equations of signature $(r,r,p)$
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by Nicolas Billerey, Imin Chen, Luis Dieulefait and Nuno Freitas PDF

Trans. Amer. Math. Soc. 371 (2019), 8651-8677 Request permission

Abstract:

In this paper, we give a resolution of the generalized Fermat equations \begin{equation*} x^5 + y^5 = 3 z^n\quad \text {and}\quad x^{13} + y^{13} = 3 z^n, \end{equation*} 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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