On Modular Approach to Diophantine Equation x⁴-y⁴=nz^p over Number Fields

2023/11/17 by Isik, Erman
#11D41 #11F03 #11F75 #11F80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.12044

Abstract

Recent results of Freitas, Kraus, Sengun and Siksek give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over various number fields. In this paper, we prove asymptotic results about the solutions of the Diophantine equation x⁴-y⁴=nz^p over various number fields using the modular method. For instance, we prove that the asymptotic generalised Fermat Theorem for the equation x⁴-y⁴=2^αz^p holds for infinitely many quadratic number fields.

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