Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers

Objectives: The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). Methods: An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell’s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan’s conjecture, binomial expansion, and various theories concerning congruence are employed. Findings: Here it is proved that, for five different choices of sides, an Exponential Diophantine triangle over can be constructed. In particular, infinitely many such triangles can be found. For some particular choice of sides, Python coding is displayed along with its output to verify the existence of required triangles. On the other side, another five different choices of sides are considered and it is shown that no considered type of triangles exists in these cases. Novelty: The idea of solving an exponential Diophantine equation and the idea of constructing triangles under some conditions using Diophantine equations already exists in the mathematical society. This article is created uniquely by combining these two concepts along with the innovative usage of exponential Diophantine equations. Keywords: Exponential Diophantine triangle, Exponential Diophantine Triangle over triangular numbers, Exponential Diophantine equation, Triangles, Triangular numbers