An Analysis on the Ternary Cubic Diophantine Equation

Abstract

In this article, we concentrate on identifying all the non-zero, infinitely many integral solutions to the ternary cubic equation 2(l^2+m^2 )-3lm=56t^3. Of these solutions, some exciting patterns are discussed.

Introduction

I. INTRODUCTION

The universal language of the world is mathematics, which imparts knowledge of numbers, structures, formulas and shapes. Integers and integral valued functions are studied in the branch of pure mathematics known as Number theory.   A polynomial equation with at least two unknowns that has only integer solutions is known as a Diophantine equation. The term ”Diophantine” refers to Diophantus of Alexandria, a third-century Hellenistic mathematician who studied these equations and was one of the first to introduce symbolism to algebra. Number theory is discussed in [3, 4, 9, 11] whereas in [6] Quadratic Diophantine equation is analysed. In [1, 2, 5, 7, 8, 10], the authors have considered cubic equation for study. In this work, a non homogeneous ternary cubic equation with three unknowns 2l2+m2-3lm=56t3 is considered in order to find some of its interesting integral solutions.

Conclusion

In this article, we have made an effort to obtain the integral solution of the non-homogeneous ternary cubic equation. Furthermore, one may search for another pattern of integral solution for the considered equation.

References

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Copyright

Copyright © 2023 G. Janaki, S. Shanmuga Priya. 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.