On the Ternary Cubic Diophantine Equation 𝑥3 + 𝑦3 = 2(𝑧 + 𝑤)2(𝑧 − 𝑤)

The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. Objectives: The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. Method: Solving Diophantine equation is obtained by the method of Decomposition. The structure of decomposition: like , where and Z. By the decomposing method in primary terms of a, we achieve a countable number of decompositions in k full factors . Each decomposition of this kind leads to a system of equations similar to: , . We get multitude of solutions for a given equation, by determining the system of equations. Findings: By the method of linear transformations, the ternary cubic equation with four unknowns is solved for its integral solutions. The equation is researched for its attributes and correlation among the solutions for its non – zero unique integer points. In each of the transformations taken, the cubic equation yields different solutions. The properties of the solutions and their relationship with the special numbers are also exhibited. Novelty: Mathematician’s interest towards solving Pell’s equation has been so much not because they approximate with a value for . The main importance of the Pell’s equation is due to that most of the common questions have answers in this equation which can be sorted by 2 variables in the Quadratic equations. This document is about the research on higher degree Cubic Diophantine equation which gives the integral solutions of this equation, taken into consideration. Keywords: Integral solutions, Ternary Cubic, Oblong number, Polygonal number