Two results on $x^r + y^r = dz^p$

Nuno Freitas; Filip Najman

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arXiv · 2207.05585

Two results on $x^r + y^r = dz^p$

Nuno Freitas·Filip Najman

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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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Nuno Freitas, Filip Najman. 2022-07-12. Two results on $x^r + y^r = dz^p$. https://arxiv.org/abs/2207.05585

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