Parametric Solutions to (six) n th Powers Equal to Another (six) n th Powers for Degree 'n' = 2,3,4,5,6,7,8,& 9

Abstract

Consider the below mentioned equation: [aann + bbnn + ccnn + ddnn + eenn + ffnn] = [ppnn + qqnn + rrnn + ssnn + ttnn + uunn]----(A). Historically in math literature there are instances where solutions have been arrived at by different authors for equation (A) above. Ref.no. (1) by A. Bremner & J. Delorme and Ref. no. (10) by Tito Piezas. The difference is that this article has done systematic analysis of equation (A) for n=2,3,4,5,6,7,8 & 9. While numerical solutions for equation (A) is available on “Wolfram math” website, search for parametric solutions to equation (A) in various publications for all n=2,3,4,5,6,7,8 & 9 did not yield much success. The authors of this paper have selected six terms on each side of equation (A) since the difficulty of the problem increases every time a term is deleted on each side of equation (A). The authors have provided parametric solutions for equation (A) for n=2, 3, 4, 5 & 6 and for n=7, 8 & 9 solutions using elliptical curve theory has been provided. Also we would like to mention that solutions for n=7, 8 & 9 have infinite numerical solutions.