Advances in Nonlinear Variational Inequalities ISSN: 1092-910X Vol 27 No. 2 (2024) 328 https://internationalpubls.com On Solving Cubic Equation   +  = ( − )  ( + ) Dr. P. Jamuna Devi 1 , Dr. K. S. Araththi 2 1 Assistant Professor, PG & Research Department of Mathematics, A.D.M College for Women (Autonomous), Nagapattinam, TamilNadu, India. Email: pjamunadevi@gmail.com 2 Assistant Professor, Department of Mathematics, MNM Jain Engineering College , Chennai, TamilNadu, India. Email: ksaraththi@gmail.com Article History: Received: 29-01-2024 Revised: 25-03-2024 Accepted: 27-04-2024 Abstract: The cubic Quadratic Equation 3 +  3 = 7( − ) 2 ( + ) is analyzed for its non-zero distinct integer solutions. Five different patterns of non-zero distinct integer solutions to the equation under consideration are obtained. A few applications of Diophantine equations are also presented. Keywords: Integral solutions, cubic Diophantine 1. Introduction The cubic Diophantine (homogeneous or non-homogeneous) equation offer an unlimited field for research due to their variety. Interesting methods like brute force methods and substitution strategies are used by few authors to solve cubic equations [1]. Some interesting results like obtaining Pythagorean triples using continued fractions through which one can solve Diophantine equation [2]. In particular,one may refer [3-18] for non-homogeneous cubic equations, with three and four unknowns. This communication concerns with yet another interesting homogenous cubic equation with four unknowns given. A few applications are also presented. Lang was a prolific mathematician known for his work in algebra, number theory, and analysis. His book "Algebra" is a classic text that covers a wide range of topics in abstract algebra, including polynomial equations. While "Algebra" may not delve deeply into homogeneous cubic equations with three unknowns specifically, it provides a solid foundation in algebraic structures and methods that are relevant to understanding and solving such equations. Artin is another influential mathematician whose contributions span algebraic geometry, number theory, and group theory. His textbook "Algebra" is widely used in undergraduate and graduate courses. While it may not focus extensively on homogeneous cubic equations with three unknowns, it covers important algebraic concepts and techniques that are applicable to solving polynomial equations of various degrees. Dummit and Foote co- authored the textbook "Abstract Algebra," which is widely used in undergraduate algebra courses. This comprehensive text covers a broad range of topics in abstract algebra, including polynomial equations, group theory, ring theory, and field theory. While it may not specifically address homogeneous cubic equations with three unknowns, it provides a thorough introduction to the algebraic structures and techniques needed to understand and potentially solve such equations.