On Certain Relations of Powers \((x^2+y^2+z^2+d^2)^r = k(ax^2+bx+c)^s(u^2+v^2+w^2)\)
Lao Hussein Mude *
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
Zachary Kaunda Kayiita
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
John Wanjau Mutuguta
Department of Mathematics and Actuarial Science, Murang’a University of Technology, P.O. Box 75-10200, Murang’a, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Let x,y, z,d,a,b,c,u,v,w and I be integers and suppose that k and r are non negative exponent. In this current study, we examine a diophantine equation relating a power relationship involving sum of four squares raised to power r and product of power of quadratic term and sums of three squares. In particular, this study, develops and introduces the diophantine equation I = (x2+y2+z2+d2)r = k(ax2+bx+c)s(u2+v2+w2), particularly when (r, s,k) = (3,1,1). This investigations involves determinations of the unknowns a,b,c,u,v and w for which the title equation has solution. The methodology involves transforming the given equation into under determine system of equations and solving via analytical method. Moreover, the study provides conjecture for the title equation.
Keywords: Diophantine Equations, Sum of Powers, polynomial equations, Title Equation