On Equal Sums of Sixth Powers and Products of Sums of Squares: \((x^6+y^6+z^6+d^6)^k = (x^2+y^2+z^2+d^2)^k(u^2+v^2+w^2)\)
Lao Hussein Mude *
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
Kinyanjui Jeremiah Ndung’u
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
Robert Muriungi Gitunga
Department of Mathematics, Meru University of Science and Technology, P. O. Box 972-60200,, Meru, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Let x,y, z,d,u,v and w be integers. This research develops and introduces the Diophantine equation (x6 +y6 +z6 +d6)k = (x2 +y2 +z2 +d2)k(u2 +v2 +w2). The equation introduces a power relationship involving sum of four sixth powers and product of sums four and three squares, providing a unique perspective on the interplay between different powers and product of sums of squares. The study involves elementary methodology grounded in integer decomposition and, factorization taking a case-by-case basis alongside generalizations. In this study the case when k = 1 has been fully determined. In particular, this study has fully generalized the case (x6 +y6 +z6 +d6) = (x2 +y2 +z2 +d2)(u2 +v2 +w2) for which z−y = y−x = d.
Keywords: Sum of sixth powers, product of sums of squares