On Sums of Squares Involving Integer Sequence: \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\)

Mude, Lao Hussein and Ndung’u, Kinyanjui Jeremiah and Kayiita, Zachary Kaunda (2024) On Sums of Squares Involving Integer Sequence: \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\). Journal of Advances in Mathematics and Computer Science, 39 (7). pp. 1-6. ISSN 2456-9968

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Abstract

Let wr be a given integer sequence in arithmetic progression with a common difference d. The study of diophantine equations, 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 squared integers from specific sequences to a variable d: In particular, the diophantine equation  \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\) is introduced and partially characterized. The objective is to determine the conditions under which integer solutions for (wr,d) exist within this diophantine equation.The methodology of solving this problem entails, decomposing polynomials, factorizing polynomials, and exploring the solution set of the given equation.

Item Type: Article
Subjects: OA Digital Library > Mathematical Science
Depositing User: Unnamed user with email support@oadigitallib.org
Date Deposited: 19 Jun 2024 05:57
Last Modified: 19 Jun 2024 05:57
URI: http://library.thepustakas.com/id/eprint/1868

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