Elementary Results on Diophantine D (n)-Triples

           In this paper, the fundamental properties of Diophantine triples with property D (n) have been investigated. The necessary and sufficient conditions for the existence of a Diophantine D (n)-triple were discussed corresponding to both positive and negative values of n. A polynomial family of Diophantine D (1)-triples, namely (k, k+2, 4k+4), was established and verified. This family provides infinitely many examples of Diophantine triples and illustrates the existence of regular algebraic patterns within the theory of Diophantine tuples. In addition, certain exceptional D (n)-triples were examined, showing that not all Diophantine triples arise from simple polynomial constructions. The results presented in this study emphasize the rich arithmetic structure of Diophantine D (n)-triples and their close relationship with perfect squares and quadratic Diophantine equations. Future research may focus on computational generation of new families with examples, extensions of triples to quadruples, and the study of connections with Pell-type equations and other topics in number theory.

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