2-Peble Triangles Over Figurate Numbers

Abstract

Objectives:The main aim of this article is to discuss the existence or nonexistence of 2-Peble triangles over some figurate numbers. Methods: A few quartic equations over integers are solved to complete the objective at hand. This is done with the aid of the transformation of variables. Additionally, fundamental concepts such as mathematical induction and parity of integers are used. Findings: Here, it is demonstrated that there are no 2-Peble triangles over triangular, hexagonal, and octagonal numbers. The same process is explained for particular special numbers as an exceptional instance. Novelty: This article defines a triangle, the d-Peble triangle over figurate numbers, which creates a link between the Pell equation and a common geometric shape. So many previous researchers, when examining a problem involving geometric shapes, attain their expected result using Diophantine equations. But this concept differs from those as this uses figurate numbers and a Pell equation to create a triangle.