Diophantine equation of degree sixteen

TL;DR

This paper explores the complex degree sixteen Diophantine equation, providing identities, parametric solutions for related equations, and numerical solutions highlighting the high degree's computational challenges.

Contribution

It introduces new identities and parametric solutions for degree four, eight, and sixteen Diophantine equations, expanding the understanding of their solutions.

Findings

New identities related to degree four and eight equations

Parametric solutions for a specific degree sixteen equation

Numerical solutions with large minimal integer values

Abstract

While there is not much publications, about degree sixteen Diophantine equation we do have an identity given by Ramanujan (ref. #1). Also on the internet even though there are numerical solutions to degree sixteen for eg. (16-7-24) equation (ref. #5) there are hardly any parametric solutions. An Octic degree parameterization has been arrived at by Choudhry & Zagar (ref. 2). The authors have given a parametric solution to the equation (a^4-b^4)(c^4-d^4)(e^8-f^8)=(u^4-v^4)(w^4-x^4)(y^8-z^8). We have also given numerical solution but because of the high degree (sixteen) of the equation we only get a minimum integer value for the variables at more than five digits. We have also given some new identities related to degree four & eight.