On Certain Relation of Sums of Squares and Quartic $$\sum_{r=1}^{2 k} a_r^4+k d^4=2 \sum_{r=1}^{2 k-1}\left(a_r a_{r+1}+d^2\right)^2$$
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.
Kinyanjui Jeremiah Ndung’u
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Let k and d be a given positive integers and suppose that ar is a given sequence. In this current study, we investiagate a diophantine identity relating the sums of squares and quartic from specific sequences to a variable d. In particular, the diophantine identity \(\sum_{r=1}^{2 k} a_r^4+k d^4=2 \sum_{r=1}^{2 k-1}\left(a_r a_{r+1}+d^2\right)^2\) is developed and introduced. The objective of this research is to determine the conditions under which integer solutions for (ar,d) exist within this diophantine equation. The methodology involves, decomposing polynomials, factorizing polynomials, and exploring the solution set of the given equation.
Keywords: Diophantine equation, sums of squares, integer sequence, quartic