Computations of Exponential Diophantine Rectangles over Gnomonic Numbers using Python

Objective: The main objective of this paper is to define and collect a new type of rectangle called the Exponential Diophantine Rectangle over Gnomonic numbers (figurate numbers that take the form 𝑛2 − (𝑛 − 1)2, 𝑛 ∈ 𝑁). Methods: It is done by solving the two exponential Diophantine equations using Mihailescu’s theorem, binomial expansion, and the basic theory of congruences. Findings: Here, it is proven that there are only four exponential Diophantine rectangles over Gnomonic numbers. Finally, it is validated using Python programming for a specific limit. Novelty: The concept of solving an exponential Diophantine equation, as well as creating rectangles under certain conditions using Diophantine equations, is already known in mathematics. But in this article, we connect the shapes and Diophantine equation which is known as Exponential Diophantine rectangles over Gnomonic numbers. 2020 MSC Classification: 11A07, 11D61, 11D72. Keywords: Exponential Diophantine Equation; Exponential Diophantine Rectangles; Mihailescu’s Theorem; Gnomonic numbers; Rectangle; Exponential Diophantine Triangles