Solution of certain Diophantine equations in Gaussian integers

2024/09/30 by Ghosh, Arkabarata
Computer Science · Mathematics · Physics and Astronomy · #11D25 #11G05 #14G05 #Advanced Mathematical Theories and Applications #Algebraic Geometry and Number Theory #Chaos-based Image/Signal Encryption #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.20416

openalex publication_date 2024/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

In this article, we show that the quartic Diophantine equations x⁴ ± pqy⁴=± z² and x⁴ ± pq y⁴= ± iz² have only trivial solutions for some primes p and q satisfying conditions p ≡ 3 \pmod 8, ~ q ≡ 1 \pmod 8 ~and~ \legendrepq = -1. Here we have found the torsion of the two families of elliptic curves to find the solutions of given Diophantine equations. Moreover, we also calculate the rank of these two families of elliptic curves over the Gaussian field ℚ(i).

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