Non-trivial Integer Solutions of $x^r+y^r=Dz^p$

TL;DR

This work studies the generalized Fermat equation $x^r+y^r=Dz^p$ with fixed $r\ge 5$ prime and coprime $D$, aiming to rule out nontrivial primitive solutions for sufficiently large $p$ within the modular method framework over totally real fields. It constructs Frey curves over $K={\mathbb Q}(\zeta_r+\zeta_r^{-1})$, proves modularity and a level-lowering mechanism, and leverages the Eichler–Shimura correspondence alongside the Weak Frey–Mazur conjecture to reduce potential solutions to finitely many elliptic curves, whose $j$-invariants can be excluded for large $p$. In the case $r\equiv 3\pmod{4}$ the unconditional arguments suffice, while for $r\equiv 1\pmod{4}$ an extra assumption $r|z$ (and inertia considerations) enables the Eichler–Shimura step; with Eichler–Shimura also assumed, a stronger conditional result follows. Overall, the paper extends the modular approach to generalized Fermat equations over totally real fields, conditional on deep conjectures, and provides ineffective bounds separating possible large exponents from existence of primitive solutions.

Abstract

In this paper, we use the modular method over totally real fields together with some standard conjectures (the Weak Frey--Mazur Conjecture and the Eichler--Shimura Conjecture) to prove that infinitely many equations of the type $x^r+y^r=Dz^p$ do not have any non-trivial primitive integer solutions, where $r \geq 5$ is a fixed prime, whenever $p$ is large enough. For $r \equiv 3 \pmod 4$, we get the same result with only assuming the Weak Frey--Mazur Conjecture.

Theorems & Definitions (39)

• Theorem 1.1: First Main Theorem
• Remark 1.2
• Theorem 1.3: Second Main Theorem
• Remark 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof