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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2019

A MULTI-FREY APPROACH TO FERMAT EQUATIONS OF SIGNATURE (r, r, p)

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In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ 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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hal-04637855 , version 1 (07-07-2024)

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Nicolas Billerey, Imin Chen, Luis Dieulefait, Nuno Freitas. A MULTI-FREY APPROACH TO FERMAT EQUATIONS OF SIGNATURE (r, r, p). Transactions of the American Mathematical Society, 2019, 371 (12), pp.8651-8677. ⟨10.1090/tran/7477⟩. ⟨hal-04637855⟩
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