On Extension of Existing Results on the Diophantine Equation:∑nr=1w2r+n3d2= 3(nd23+∑n3r=1w23r−1)

Abstract

Let $w_r$ be a given sequence in arithmetic progression with common difference $d$. The study of diophantine equation, which are polynomial equations seeking integer solutions has been a very interesting journey in the field of number theory. Historically, these equations have attracted the attention of many mathematicians due to their intrinsic challenges and their significance in understanding the properties of integers. In this current study we examine a diophantine equation relating the sum of square integers from specific sequences to a variable $d$. In particular, on extension of existing results on the diophantine equation: $\sum_{r=1}^{n} w^2_r +\frac{n}{3}d^2= 3(\frac{nd^2}{3} +\sum^{\frac{n}{3}}_{n=1} w^{2}_{3r-1})$ is introduced and partially characterized.