2-Peble Triangles Over Figurate Numbers

M Mahalakshmi; J Kannan; A Deepshika; K Kaleeswari

doi:10.17485/IJST/v16i44.2663

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Indian Journal of Science and Technology

DOI: 10.17485/IJST/v16i44.2663

Year: 2023, Volume: 16, Issue: 44, Pages: 4108-4113

Original Article

2-Peble Triangles Over Figurate Numbers

M Mahalakshmi¹, J Kannan^2*, A Deepshika³, K Kaleeswari¹

¹Ph.D. Scholar, Department of Mathematics, Ayya Nadar Janaki Ammal College (Autonomous, affiliated to Madurai Kamaraj University), Sivakasi, India
²Assistant Professor, Department of Mathematics, Ayya Nadar Janaki Ammal College (Autonomous, affiliated to Madurai Kamaraj University), Sivakasi, India
³Ph.D. Scholar, Department of Mathematics, Ayya Nadar Janaki Ammal College (Autonomous, affiliated to Madurai Kamaraj University), Sivakasi, India

*Corresponding Author
Email: [email protected]

Received Date:20 October 2023, Accepted Date:28 October 2023, Published Date:28 November 2023

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Objectives: The main aim of this article is to discuss the existence or non-existence of -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.

Keywords: 2-Peble Triangle, Figurate Numbers, Pell Equations, Exponential Diophantine Triangle, Quartic Equation

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Copyright

© 2023 Mahalakshmi et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)