Investigating Computational Generalization of Mixed Polynomial Exponential on Diophantine Equations: αn + βn + α(αsψβs)m + D = r(uk + vk + wk) with consecutive α and β

Abraham Osogo Nyakebogo; Boaz Simatwo Kimtai

doi:10.51867/Asarev.Maths.2.1.1

Authors

Abraham Osogo Nyakebogo Department of Mathematics and Actuarial Science, Kisii University, P. O. Box 408-40200, Kisii, Kenya Author https://orcid.org/0000-0001-5683-1469
Boaz Simatwo Kimtai Department of Mathematics, Masinde Muliro University of Science and Technology, P.O. Box 190-50100, Kakamega, Kenya Author https://orcid.org/0000-0002-1314-6119

DOI:

https://doi.org/10.51867/Asarev.Maths.2.1.1

Keywords:

Sequences, Diophantine equation, Integers, Polynomial identities, Factorization

Abstract

Let a, α, β, r, u, v, w, and D be integers, and let n, m, s, and k be non-negative exponents. This paper investigates the Diophantine equation αⁿ + βⁿ + a(αˢ + βˢ)ᵐ + D = r(uᵏ + vᵏ + wᵏ) for integer solutions and explores associated polynomial identities. Additionally, several conjectures are formulated in relation to this equation, aiming to extend the understanding of its integer solutions and structural properties. The focus is on finding integer solutions and examining the underlying polynomial identities. The structure of this equation—combining additive powers, multiplicative transformations, and polynomial expressions in multiple integer variables—presents unique challenges and opportunities in the search for integer solutions. By exploring cases for specific values of n, m, s, and k, we develop insights into solution sets, identify relationships among terms, and analyze the symmetry properties inherent in this equation.

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 DOI: 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 DOI: 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.

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 DOI: 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 cube. Science Mundi, 3(1), 135-142.

https://doi.org/10.51867/scimundi.3.1.14 DOI: 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. https://doi.org/10.21203/rs.3.rs-882192/v1 DOI: https://doi.org/10.21203/rs.3.rs-882192/v1

Lao, H. M., Zachary, K. K., & Kinyanjui, J. N. (2023). Some generalized formula for sums of cube.

Mochimaru, Y. (2005). Solution of sextic equations. International Journal of Pure and Applied Mathematics, 23(4), 577.

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/v37i430448 DOI: https://doi.org/10.9734/jamcs/2022/v37i430448

Mude, L. H. (2024). On some mixed polynomial exponential Diophantine equation: αⁿ + βⁿ + a(αˢ ± βˢ)^m + d = r(uᵏ + vᵏ + wᵏ) with α and β consecutive. Journal of Advances in Mathematics and Computer Science, 39(10), 11-17.

https://doi.org/10.9734/jamcs/2024/v39i101931 DOI: 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: ∑₍ᵣ₌₁₎ⁿ wᵣ² + (n/3)d² = 3[(nd²/3) + ∑₍ᵣ₌₁₎ⁿ⁄₃ w₍₃ᵣ₋₁₎²]. Journal of Advances in Mathematics and Computer Science, 39(7), 1-6.

https://doi.org/10.9734/jamcs/2024/v39i71906 DOI: https://doi.org/10.9734/jamcs/2024/v39i71906

Najman, F. (2010a). The Diophantine equation x⁴ ± y⁴ = iz² in Gaussian integers. The American Mathematical Monthly, 117(7), 637-641. https://doi.org/10.4169/000298910x496769 DOI: 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/10.1016/j.jnt.2009.12.008 DOI: https://doi.org/10.1016/j.jnt.2009.12.008

Obiero, B. A., & Simatwo, K. B. (2025). On certain results on the Diophantine equation: ∑₍ᵣ₌₁₎ⁿ wᵣ² + (n/3)d² = 3[(nd²/3) + ∑₍ᵣ₌₁₎ⁿ⁄₃ w₍₃ᵣ₋₁₎²]. Journal of Advances in Mathematics and Computer Science, 40(2), 1-7. https://doi.org/10.9734/jamcs/2025/v40i21966 DOI: https://doi.org/10.9734/jamcs/2025/v40i21966

Osogo, N. A., & Simatwo, K. B. (2025). On extension of existing results on the Diophantine equation: ∑₍ᵣ₌₁₎ⁿ wᵣ² + (n/3)d² = 3[(nd²/3) + ∑₍ᵣ₌₁₎ⁿ⁄₃ w₍₃ᵣ₋₁₎²]. Earthline Journal of Mathematical Sciences, 15(2), 201-209. https://doi.org/10.34198/ejms.15225.201209 DOI: https://doi.org/10.34198/ejms.15225.201209

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 DOI: 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 DOI: https://doi.org/10.1142/9719