Homogeneous Quadratic Equation with Four Unknowns 𝑥2 + 𝑥𝑦 + 𝑦2 = 𝑧2 + 𝑧𝑤 + 𝑤2

R Sathiyapriya  lowast; M A Gopalan

doi:10.17485/IJST/v17i27.1710

Article

Homogeneous Quadratic Equation with Four Unknowns 𝑥2 + 𝑥𝑦 + 𝑦2 = 𝑧2 + 𝑧𝑤 + 𝑤2

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Indian Journal of Science and Technology

DOI: 10.17485/IJST/v17i27.1710

Year: 2024, Volume: 17, Issue: 27, Pages: 2841-2847

Original Article

Homogeneous Quadratic Equation with Four Unknowns 𝑥2 + 𝑥𝑦 + 𝑦2 = 𝑧2 + 𝑧𝑤 + 𝑤2

R Sathiyapriya^1∗, M A Gopalan²

¹Associate Professor, Department of Mathematics, School of Engineering and Technology, Dhanalakshmi Srinivasan University, Trichy, 621112, Tamil Nadu, India
²Professor, Department of Mathematics, Shrimati Indira Gandhi College, (Affiliated to Bharathidasan University), Trichy, 620002, Tamil Nadu, India

*Corresponding Author
Email: [email protected]
[email protected]

Received Date:18 May 2024, Accepted Date:17 June 2024, Published Date:16 July 2024

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Objectives: Diophantine research focuses on various ways to tackle multi variable and multi-degree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to Polynomial Diophantine equation . Methods: Diophantine equations may have finite, infinite, or no solutions in integers. There is no universal method for finding solutions to Diophantine equations. Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Findings: Many distinct patterns of integer solutions are obtained. Novelty: The main thrust is to illustrate different ways of obtaining various choices of solutions in integers to second-degree equations with four variables . Different choice of solutions in integers is obtained through using linear transformations and employing the factorization method. Utilization of substitution strategy reduces the given equation to a ternary quadratic equation for which solutions can be found easily. Mathematics Subject Classification:11D09

Keywords: Homogeneous second degree with four variables, Solutions in integers, Factorization method, Linear transformation, Polynomial diophantine equation

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Copyright

© 2024 Sathiyapriya & Gopalan. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)