Two results on 𝑥^{𝑟}+𝑦^{𝑟}=𝑑𝑧^{𝑝}

Freitas, Nuno; Najman, Filip

doi:10.1090/proc/16575

Two results on $x^r + y^r = dz^p$
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by Nuno Freitas and Filip Najman;

Proc. Amer. Math. Soc. 152 (2024), 591-598

DOI: https://doi.org/10.1090/proc/16575

Published electronically: November 7, 2023

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Abstract:

This note proves two theorems regarding Fermat-type equation $x^r + y^r = dz^p$ where $r \geq 5$ is a prime. Our main result shows that, for infinitely many integers $d$, the previous equation has no non-trivial primitive solutions such that $2 \mid x+y$ or $r \mid x+y$, for a set of exponents $p$ of positive density. We use the modular method with a symplectic argument to prove this result.

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