Non-trivial Solutions of Aa^p+Bb^p=Cc³ over Number Fields

2025/10/15 by Kara, Yasemin, Nomden, Stef, Özman, Ekin
#11D41 #11F75 #11F80 #11G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.16029

Abstract

In this paper, we investigate solutions to the Diophantine equation A a^p + B b^p = C c³ 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 = c³ over K = ℚ(√(-d)) with d ∈ \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.

Citations

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• On Ternary Diophantine Equations of Signature (p,p,3) over Number Fields
• Class field theory, Diophantine analysis and the asymptotic Fermat's\n Last Theorem
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• Sums of two cubes as twisted perfect powers, revisited