The Fermat-type equations 𝑥⁵+𝑦⁵=2𝑧^{𝑝} or 3𝑧^{𝑝} solved through ℚ-curves

Dieulefait, Luis; Freitas, Nuno

doi:10.1090/S0025-5718-2013-02731-7

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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