On integer solutions of x⁴ + y⁴ – 2z⁴ – 2w⁴ = 0

Dustin Moody and Arman Shamsi Zargar
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 1, Pages 37–43
Full paper (PDF, 164 Kb)

Details

Authors and affiliations

Dustin Moody
Computer Security Division, NIST
100 Bureau Drive, Gaitherburg, MD, 20899-8930

Arman Shamsi Zargar
Department of Mathematics, Azarbaijan Shahid Madani University
Tabriz, Iran

Abstract

In this article, we study the quartic Diophantine equation x⁴ + y⁴ – 2z⁴ – 2w⁴ = 0. We find non-trivial integer solutions. Furthermore, we show that when a solution has been found, a series of other solutions can be derived. We do so using two different techniques. The first is a geometric method due to Richmond, while the second involves elliptic curves.

Keywords

• Diophantine equation
• Congruent elliptic curve

AMS Classification

• 11G05

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