On the Diophatine Equation \( a b(c d+1)+L=u^{2}+v^{2}\)
Lao Hussein Mude *
Department of Pure and Applied Sciences, Kirinyaga University , P. O. Box 143-10300, Kerugoya, Kenya.
*Author to whom correspondence should be addressed.
Abstract
The search for integer sums of two squares is still an open problem. A number of research conducted so far has put more attention on Fermat sums of two squares with little attention given to other methods of generating sums of two squares. Let a; b; c; d; u; v and L be positive integers. This study introduces the relation \( a b(c d+1)+L=u^{2}+v^{2}\)for generating integer sums of two squares. The main objective of this study is to develop general formulae by determining the integer values a; b; c; d; u; v and L such that\( a b(c d+1)+L=u^{2}+v^{2}\) . Moreover, this research provides conjecture for the title equation.
Keywords: Diophantine equation, sums of two squares