• P-ISSN 0974-6846 E-ISSN 0974-5645

# Indian Journal of Science and Technology

## Article

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Indian Journal of Science and Technology

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

Original Article

## 2-Peble Triangles Over Figurate Numbers

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

## 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

## References

1. Ddamulira M. On the x--coordinates of Pell equations that are products of two Lucas numbers. The Fibonacci Quarterly. 2020;58(1):18–37. Available from: https://hal.science/hal-02266433/document
2. Erazo HS, Gómez CA, Luca F. On Pillai's problem with X-coordinates of Pell equations and powers of 2. Journal of Number Theory. 2019;203:294–309. Available from: https://doi.org/10.1016/j.jnt.2019.03.010
3. Kafle B, Luca F, Montejano A, Szalay L, Togbé A. On the X-coordinates of Pell equations which are products of two Fibonacci numbers. Journal of Number Theory. 2019;203:310–333. Available from: https://doi.org/10.1016/j.jnt.2019.03.011
4. Mahalakshmi M, Kannan J. Some annotations on almost and pseudo almost equilateral rational rectangles. Research and Reflections on Education. 2022;20(3A):88–93. Available from: https://www.sxcejournal.com/spe-oct-2022/15.pdf
5. Mahalakshmi M, Kannan J, Narasimman G. Certain sequels on almost equilateral triangles. Advances and Applications in Mathematical Sciences. 2022;22(1):149–157. Available from: https://www.mililink.com/upload/article/2146637988aams_vol_221_november_2022_a13_p149-157_m._mahalakshmi_et_al..pdf
6. Kannan J, Somanath M, Raja K. On the class of solutions for the hyperbolic Diophantine equation. International Journal of Apllied Mathematics. 2019;32(3):443–449. Available from: https://doi.org/10.12732/ijam.v32i3.6
7. Somanath M, Raja K, Kannan J, Mahalakshmi M. On a class of solution for a quadratic Diophantine equation. Advances and Applications in Mathematical Sciences. 2020;19(11):1097–1103. Available from: https://www.mililink.com/upload/article/988291244aams_vol_1911_sep_2020_a2_p1097-1103__manju_somanath_and_m._mahalakshmi.pdf
8. Kannan J, Somanath M. Fundamental Perceptions in Contemporary Number Theory. (pp. 213-226) New York. Nova Science Publishers. 2023.
9. Keskin R, Duman MG. Positive integer solutions of some Pell equations. Palestine Journal of Mathematics. 2019;8(2):213–239. Available from: https://pjm.ppu.edu/sites/default/files/papers/PJM_April2019_213to226.pdf
10. Andreescu T, Andrica D, Cucurezeanu I. An introduction to Diophantine equations: A problem-based approach. New York. Birkhauser. 2010.