Intrinsic Solution on Ternary Quadratic Diophantine Equation

Abstract

Numerous ternary quadratic equations exist. For a more complete understanding, check [1-7]. The And form of the ternary quadratic Diophantine problem has been studied for solving non-trivial integrals. In [9-15], non-zero integral solutions to the various Diophantine equations are investigated. These findings have motivated us to hunt for an infinite number of non-zero integral solutions to the ternary quadratic equation given by, which is another fascinating equation. There are also a few intriguing correlations between the answers, as well as some unique numbers such as triangular, cantered, gnomon, and star. In addition, a Python program is used to code in the quadratic diophantine equation in the five patterns to determine the program\'s output.

Introduction

The study focuses on solving ternary quadratic Diophantine equations, which have numerous forms and applications, including solving non-trivial integrals. Prior works ([1-15]) have explored non-zero integral solutions for various such equations, inspiring the search for infinitely many non-zero integral solutions to a specific ternary quadratic equation. The solutions relate interestingly to special number types like triangular, centered hexagonal, gnomon, and star numbers. A Python program is also implemented to solve the equation in five distinct pattern templates.

Key points:

• Notations: Definitions of special numbers used (Triangular, Gnomonic, Star, Centered Hexagonal).

• Methodology: The problem is approached by expressing the ternary quadratic Diophantine equation in a standard form and applying linear transformations and factorization methods.

• Five Templates: Different algebraic manipulations and substitutions generate families of non-zero integral solutions.

  • Each template involves rewriting the equation, applying transformations, factorization, and solving for integer values.

  • Observations are made on conditions to choose parameters that yield valid integer solutions.

• Computational Aspect: A Python program is developed to automate the solving process for these five templates and visualize results via line graphs.

Conclusion

This work presents infinitely many non-zero unique integer solutions to the ternary quadratic diophantine problem , along with some observations about the solutions. For alternative options of ternary quadratic diophantine equations, one could look for additional examples of non-zero whole number remarkable arrangements and their corresponding highlights, or increase the quadratic diophantine equation of five patterns using Python program coding of line graph to find a diagram output.

References

[1] Carmichael, R.D., The theory of numbers and Diophantine Analysis, Dover Publications,New York, 1959. [2] Batta. B and Singh. A.N, History of Hindu Mathematics, Asia Publishing House 1938. [3] Dickson L.E, History of Theory of Numbers, Vol.11, Chelsea Publishing company, New York,1952. [4] Mordell. L.J, Diophantine equations, Academic Press,London,1969 Telang, S.G., Number theory, Tata McGraw Hill publishing company, New Delhi, 1996. [5] Janaki.G and Saranya.C., Observations on the Ternary Quadratic Diophantine Equation . International Journal of Innovative Research in Science, Engineering and Technology, Vol-5, Issue-2, Pg.no: 2060-2065, Feb 2016. [6] Gopalan.M.A., Vidhyalakshmi.S and Umarani.J., “On ternary Quadratic Diophantine equation , Sch.J. Eng. Tech. 2(2A); 108- 112,2014. [7] Janaki. G, and Saranya.C, Integral solutions of the Ternary cubic equation , International Research Journal of Engineering and Technology, vol 4, issue 3, 665-669, March 2017. [8] Dr Kavitha A, Sasi Priya P, A Ternary Quadratic Diophantine Equation journal of Mathematics and Informatics, vol 11, 2017,103-109. [9] Selva Keerthana K , Mallika S, “On the ternary quadratic Diophantine equation , Journal of Mathematics and informatics, vol.11, 21-28,2017. [10] Dr.S. Mallika, D.Maheshwari, R. Anbarasi, On the homogeneous Ternary Quadratic Diophantine equation Infokara Research Journal, Volume 9, issue 4, April 2020, page 26-31. [11] Janaki, G., & Sangeetha, P. (2024). On the ternary quadratic Diophantine equation ,Arya Bhatta Journal of Mathematics and Informatics Journal,Volume 16,issue 1,page 49-54. [12] Janaki, G. and Sangeetha, P. 2023. “Integral solution of the ternary cubic equation Asian Journal of Science and Technology, 14, (05), 12509-12512. [13] R. Rasini G. Janaki, P. Sangeetha. 2024/3.” Integer Solutions on Ternary Quadratic Diophantine Equation .International Journal for Research in Applied Science & Engineering Technology. Volume 12, issue 3, page 761-765.

Copyright

Copyright © 2025 G.Janaki pavithra, P. Sangeetha, B. Pavithra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.