Algebraic structure and a study of special Diophantine and Pythagorean equations

Srinivas, T.

doi:10.47974/JDMSC-2564

Open Access Research Article

Algebraic structure and a study of special Diophantine and Pythagorean equations

^*T. SrinivasCorresponding authordrtsrinivas80@gmail.comDepartment of Freshman EngineeringAudisankara (Deemed to be University), GudurNellore, Andhra Pradesh, 524101, India0009-0001-6282-7762View full profile →

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pp. 2261–2267Vol. 29Issue 6June 2026DOI: 10.47974/JDMSC-2564 Crossmark XML

Received:: 01 May 2025
Published Online:: 16 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2564
Pages:: 2261–2267

Abstract

This research paper explores advanced Diophantine equations with Pythagorean characteristics and their algebraic properties. The work extends classical number theory concepts into novel structural frameworks, particularly through its examination of solution spaces and binary operation. The core equation kx2 + y2 = z2 transforms into normalized ellipse form of k(x/z)2+(y/z)2 = 1. The reciprocal variant k/p2 +1/q2 =1/h2 introduces new constraints of integer triples (p, q, h). Also, the paper proposes a complex binary operation P1.P1 defined through component-wise combinations: P1.P1 = {(kx1 x22, y1 (y22+z22 )+2y2 z1 z2, z1 (y22+z22) + 2y1 y2 z2 ) @ (kx2 x12, y2 (y12 + z12 ) +2y1 z1 z2, z2 (y12 + z12 ) + 2y1 y2 z1)} Also focused Extension of Binomial Distribution for kx2 + y2 = z2.

Keywords

Algebraic structure Binary operation Associative axiom

Subject Classifications

11D7211D61

References

[1] K. Sridevi and T. Srinivas, “Transcendental representation of the Diophantine equation and some of its inherent properties,” Mater. Today: Proc., vol. 80, pp. 1822–1825 (2023).
[2] T. Srinivas and K. Dhanumjaya, “Integer solutions parameterized by k, m, n yielding unified crypto-control pipelines,” Int. J. Adv. Sci. Tech. Res. (IJASTR), vol. 16, no. 1, pp. 208–220 (2026), doi: 10.5281/zenodo.18368265.
[3] K. Sridevi and T. Srinivas, “Cryptographic coding to define binary operation on a set of Pythagorean triplets,” Mater. Today: Proc., vol. 80, pp. 1822–1825 (2023).
[4] T. Srinivas, “A study on quadratic and quintic Diophantine equation in embedded cryptographic coprocessors,” Int. J. Comput. Appl. (IJCA), vol. 16, no. 1, pp. 22/1–22/12 (2026). doi: 10.26808/RS.2026.c1fb5f
[5] Siswanto and Sahmura Maula Al-Maghribi, “Solving the eigenproblems of triangular and diagonal strictly double ℝ-astic matrices over min-plus algebra,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 28, no. 6, pp. 2237–2245 (2025), doi: 10.47974/JDMSC-2188.
[6] S. Muthuvel and R. Venkatraman “An exponential Diophantine equation x^2 + 3^a*73^b = y^n,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 28, no. 6, pp. 2289–2296 (2025), DOI: 10.47974/JDMSC-2225.

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