Parametric Solution to six n degree powers for degree two, three, four, five, six , seven, eight, and nine

TL;DR

This paper systematically derives parametric solutions for a specific Diophantine equation involving six terms on each side across degrees 2 to 9, using elliptic curve methods, revealing infinite solutions for degrees 7 to 9.

Contribution

It provides the first systematic parametric solutions for degrees 2 through 9 for this equation, including elliptic curve approaches for degrees 7 to 9.

Findings

Parametric solutions for degrees 2 to 6 are explicitly derived.

Elliptic curve methods yield infinite solutions for degrees 7 to 9.

The approach advances understanding of complex Diophantine equations.

Abstract

Historically in math literature there are instances where solutions have been arrived at by different authors for equation 1 for six powers on both side of equation, for different degree 2,3,4,5,6,7,8,9. See reference number 1, by A. Bremner & J. Delorme and reference number 10, by Tito Piezas. The difference is that this paper has done systematic analysis of equation 1. While numerical solutions for equation 1, is available on Wolfram math website, search for parametric solutions to equation 1, in various publications for all degree 2,3,4,5,6,7,8,9 did not yield results. The authors of this paper have selected six terms on each side of equation 1, since the difficulty of the problem increases every time a term is deleted on each side of equation 1. The authors have provided parametric solutions for equation 1, for degree 2, 3, 4, 5 & 6 and for degree 7, 8 & 9 solutions using elliptical curve method has been provided. Also we would like to mention that solutions for degree 7, 8 & 9 using elliptic curve method has infinite numerical solutions. Keywords, Pure math, Diophantine equations, Equal sums, parameteric equations.