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arXiv · 2510.16029

Non-trivial Solutions of $Aa^p+Bb^p=Cc^3$ over Number Fields

Yasemin Kara·Stef Nomden·Ekin Özman
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Abstract

In this paper, we investigate solutions to the Diophantine equation $ A a^p + B b^p = C c^3 $ over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate $S$-unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation $a^p + d b^p = c^3$ over $ K = \mathbb{Q}(\sqrt{-d}) $ with $ d \in \{7, 19, 43, 67\} $, we determine an explicit bound (depending on $ d $) such that no solutions of a certain type exist whenever $ p $ exceeds this bound.

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Yasemin Kara, Stef Nomden, Ekin Özman. 2025-10-15. Non-trivial Solutions of $Aa^p+Bb^p=Cc^3$ over Number Fields. https://arxiv.org/abs/2510.16029

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