To identify its different integral non-zero solutions, the ternary cubic equation is taken into consideration. Different integral solution patterns to the ternary cubic equation under consideration are obtained in each pattern by using the linear transformation and the method of factorization; interesting relationships between the solutions and some polygonal numbers, such as pyramidal and central pyramidal numbers, are also displayed.
Three unique patterns of non-zero distinct integer solutions to the given non-homogeneous problem are shown in this paper.
For various options of cubic Diophantine equations, additional patterns of non-zero integer unique solutions and their corresponding characteristics may be found.
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