Derivation of Finite Number of Integer Solutions to Particular Form of Mordell’s Equation a2 = b3+r2; r = 8; 9; 10

V Pandichelvi; S Saranya

doi:10.17485/IJST/v16i28.1322

Article

Derivation of Finite Number of Integer Solutions to Particular Form of Mordell’s Equation a2 = b3+r2; r = 8; 9; 10

VIEWS 317
PDF 93

Indian Journal of Science and Technology

DOI: 10.17485/IJST/v16i28.1322

Year: 2023, Volume: 16, Issue: 28, Pages: 2113-2117

Original Article

Derivation of Finite Number of Integer Solutions to Particular Form of Mordell’s Equation a2 = b3+r2; r = 8; 9; 10

V Pandichelvi¹, S Saranya^2*

¹Assistant Professor, PG & Research Department of Mathematics, Urumu Dhanalakshmi College (Affiliated to Bharathidasan University), Trichy, India
²Research Scholar, PG & Research Department of Mathematics, Urumu Dhanalakshmi College (Affiliated to Bharathidasan University), Trichy, India

*Corresponding Author
Email: [email protected]

Received Date:31 May 2023, Accepted Date:18 June 2023, Published Date:22 July 2023

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

Abstract

Objectives: The perception of solving Diophantine equations is a massive area of research. Mordell equation is a type of Diophantine equation such that the difference of square and cube of numbers remains constant. Various authors analyses Mordell kind equations for existence of solutions by applying several methods. The objective of this manuscript is whether an equation of the form such that the difference of square and cube of numbers provides square of a particular integer. The possibility of visualization of the surface of the considered equation by using the solutions with the support of MATLAB algorithm is also investigated. Method: Diophantine equations may have infinite number of solutions or finite number of solutions or no solutions in integers. There is no universal method for solving Diophantine equations. The particular type of Mordell’s equation is scrutinized by the concept of divisibility. Findings: A special type of Mordell’s equation such that the difference of square and cube of numbers offers square of a certain integer is inspected by daubing the notions of divisibility. Furthermore, the integer solutions and the corresponding surface of the equation are recognized with the support of MATLAB programs. Novelty: In the analysis, there is no solutions for few cases. The results are confirmed by MATLAB programs. If solutions to the equation exist, the surface of an equation is envisioned with the assistance of MATLAB program.

Keywords: Mordell’s Diophantine Equation; Integer Solutions; Divisibility

References

1) Everest G, Ward T. An introduction to number theory. Springer Science & Business Media. 2006;232. Available from: https://doi.org/10.1007/b137432.

2) Flath DE. Introduction to Number Theory. American Mathematical Society. 2018. Available from: https://bookstore.ams.org/view?ProductCode=CHEL/

384.H.

3) Niven I, Hibert S, and ZH. An Introduction to the Theory of Numbers. John Wiley & Sons Inc. 2004. Available from: https://undergraduatemaths.files.

wordpress.com/2017/12/ivan_niven_herbert_s-_zuckerman_hugh_l-_montgoman-introduction-to-the-theory-of-numbers.pdf.

4) Bennett MA, Ghadermarzi A. Mordell’s equation: a classical approach. LMS Journal of Computation and Mathematics. 2015;18(1):633–646. Available

from: https://doi.org/10.1112/S1461157015000182.

5) Boyer W. Integer solutions to x^2+y^2= z^2- k for a fixed integer value k. Journal of Mathematics. 2017;10:881–892. Available from: https://doi.org/10.

2140/involve.2017.10.881.

6) Gebel J, Pethö A, Zimmer HG. On Mordell‘s equation. Compositio Mathematica. 1998;110:335–367. Available from: https://doi.org/10.1023/A:

1000281602647.

7) Känel V, Rafael, Matschke B. Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture.

2016. Available from: https://doi.org/10.1090/memo/1419.

8) Gauthier S, Lê F. On the youthful writings of Louis J. Mordell on the Diophantine equation y^2-k= x^3. Archive for History of Exact Sciences. 2019;73:427–

468. Available from: https://doi.org/10.1007/s00407-019-00231-1.

9) Mohanty SP. On consecutive integer solutions for y2-k= x3. Proceedings of the American Mathematical Society. 1975;48:281–285. Available from:

https://doi.org/10.1090/S0002-9939-1975-0357319-8.

10) Pandichelvi V, Sandhya P. Attesting finite number of integer solutions or no integer solutions to four Mordell kind Equations Y2=X3+C,C=9,36,􀀀16.

2022. Available from: https://www.mililink.com/upload/article/167889248aams_vol_218_june_2022_a43_p4693-4701_v._pandichelvi_and_vanaja.pdf.

Copyright

© 2023 Pandichelvi & Saranya. 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)