On Extension of Existing Results on the Diophantine Equation: \(\sum_{r=1}^n w_r^2+\frac{n}{3} d^2=3\left(\frac{n d^2}{3}+\sum_{r=1}^{\frac{n}{3}} w_{3 r-1}^2\right)\)
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.
References
Bombieri, E., & Bourgain, J. (2015). A problem on sums of two squares. International Mathematics Research Notices, 2015(11), 3343-3407. https://doi.org/10.1093/imrn/rnu005
Cai, Y. (2016). Waring-Goldbach problem: two squares and higher powers. Journal de Théorie des Nombres de Bordeaux, 28(3), 791-810. https://doi.org/10.5802/jtnb.964
Cavallo, A. (2019). An elementary computation of the Galois groups of symmetric sextic trinomials. arXiv preprint arXiv:1902.00965. https://doi.org/10.48550/arXiv.1902.00965
Christopher, A. D. (2016). A partition-theoretic proof of Fermat's two squares theorem. Discrete Mathematics, 339(4), 1410-1411. https://doi.org/10.1016/j.disc.2015.12.002
Fathi, A., Mobadersany, P., & Fathi, R. (2012). A simple method to solve quartic equations. Australian Journal of Basic and Applied Sciences, 6(6), 331-336.
Kimtai, B. S., & Mude, L. H. (2023). On generalized sums of six, seven, and nine cubes. Science Mundi, 3(1), 135-142. https://doi.org/10.51867/scimundi.3.1.14
Kouropoulos, G. P. (2021). A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation. Unpublished. https://doi.org/10.21203/rs.3.rs-882192/v1
Mude, L. H., Kayiita, Z. K., & Ndung'u, K. J. (2023). Some generalized formula for sums of cube. Journal of Advances in Mathematics and Computer Science, 38(8), 47-52. https://doi.org/10.9734/jamcs/2023/v38181789
Mochimaru, Y. (2005). Solution of sextic equations. International Journal of Pure and Applied Mathematics, 23(4), 577. Academic Publications.
Mude, L. H. (2022). Some formulae for integer sums of two squares. Journal of Advances in Mathematics and Computer Science, 37(4), 53-57. https://doi.org/10.9734/jamcs/2022/v371430448
Mude, L. H. (2024). On some mixed polynomial exponential Diophantine equation: $(alpha^n + beta^n + a(alpha^s pm beta^s)^m + D = r(u^k + v^k + w^k))$ with $(alpha)$ and $(beta)$ consecutive. Journal of Advances in Mathematics and Computer Science, 39(10), 11-17. https://doi.org/10.9734/jamcs/2024/v39i101931
Mude, L. H., Ndung'u, K. J., & Kayiita, Z. K. (2024). On sums of squares involving integer sequence: $left(sum_{r=1}^{n} w_r^2 + frac{n}{3} d^2 = 3left(frac{nd^2}{3} + sum_{r=1}^{n} frac{n}{3} w_{3r-1}^2right)right)$. Journal of Advances in Mathematics and Computer Science, 39(7), 1-6. https://doi.org/10.9734/jamcs/2024/v39171906
Najman, F. (2010a). The Diophantine equation ( x^4 pm y^4 = iz^2 ) in Gaussian integers. The American Mathematical Monthly, 117(7), 637-641. https://doi.org/10.4169/000298910x496769
Najman, F. (2010b). Torsion of elliptic curves over quadratic cyclotomic fields. arXiv preprint arXiv:1005.0558. https://doi.org/10.48550/arXiv.1005.0558
Simatwo, K. B. (2024). Equal sums of four even powers. Asian Research Journal of Mathematics, 20(5), 50-54. https://doi.org/10.9734/arjom/2024/v20i5802
Tignol, J.-P. (2015). Galois' theory of algebraic equations. World Scientific Publishing Company. https://doi.org/10.1142/9719
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