International Journal of Mathematics Trends and Technology
Volume 68 | Issue 12 | Year 2022 | Article Id. IJMTT-V68I12P503 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I12P503
| Received | Revised | Accepted | Published |
|---|---|---|---|
| 19 Oct 2022 | 20 Nov 2022 | 03 Dec 2022 | 15 Dec 2022 |
In this manuscript, the set of non-zero integer solutions (x,y) to some Diophantine equations of the forms x2-rxy-y2=±1 where 4≤ r≤ 9 by relating continued fraction and metallic ratios are investigated. Furthermore, the collection of all such pair (x,y) are confirmed by MATLAB programs.
[1] Niven, Ivan, Herbert S. Zuckerman, and Hugh L. Montgomery, “An Introduction to the Theory of Numbers,” John Wiley & Sons, 1991.
[2] Tom M Apostol, “Introduction to Analytic Number Theory,” Springer Science & Business Media, 1998. Crossref, https://doi.org/10.1007/978-1-4757-5579-4
[3] Jiri Herman, Radan Kucera, and Jaromir Simsa, “Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory,” Springer Science & Business Media, vol. 1, 2000. Crossref, https://doi.org/10.1007/978-1-4612-1270-6
[4] Song Y Yan, "Elementary Number Theory," Number Theory for Computing, Springer, Berlin, Heidelberg, pp. 1-172, 2002. Crossref, https://doi.org/10.1007/978-3-662-04773-6
[5] Richard K. Guy, “Unsolved Problems in Number Theory,” Springer Science & Business Media, vol. 1, 2004. Crossref, https://doi.org/10.1007/978-0-387-26677-0
[6] Graham Everest, and Thomas Ward, “An Introduction to Number Theory,” Springer Science & Business Media, vol. 232, 2006. Crossref, https://doi.org/10.1007/b137432
[7] Vazzana, Anthony, Martin Erickson, and David Garth, “Introduction to Number Theory,” Chapman and Hall/CRC, 2007.
[8] Cohen Henri, “Number Theory: Volume II: Analytic and Modern Tools,” Springer Science & Business Media, vol. 240. 2008. Crossref, https://doi.org/10.1007/978-0-387-49894-2
[9] David M. Burton, “EBOOK: Elementary Number Theory,” McGraw Hill, 2010.
[10] André Weil, “Basic Number Theory,” Springer Science & Business Media, vol. 144, 2013. Crossref, https://doi.org/10.1007/978-3-642- 61945-8
[11] Jürgen Neukirch, “Algebraic Number Theory,” Springer Science & Business Media, vol. 322, 2013. Crossref, https://doi.org/10.1007/978-3-662-03983-0
[12] Bilge Peker, "Solutions of the Pell Equations x2-(a2+2a)y2=N via Generalized Fibonacci and Lucas Numbers," arXiv preprint arXiv:1304.1043, 2013. Crossref, https://doi.org/10.48550/arXiv.1304.1043
[13] Refik Keskin, and ZaferŞiar, "Positive Integer Solutions of the Diophantine Equation x2-Lnxy+(-1)ny2=±5r," Proceedings Mathematical Sciences, vol. 124, no. 3, pp. 301-313, 2014.
[14] Refik Keskin, Olcay Karaatlı, and Zafer Siar, "Positive Integer Solutions of the Diophantine Equations x2-5Fnxy-5(-1)ny2=±5r." Miskolc Mathematical Notes, vol. 14, no.3, pp. 959-972, 2013.
[15] Gopalan, M. A., V. Sangeetha, and Manju Somanath. "On the Integer Solutions of the Pell Equation x2=13y2-3t," International Journal of Applied Mathematical Research, vol. 3, no. 1, pp. 58-61, 2014.
[16] V. Pandichelvi, P. Sivakamasundari, and M.A. Gopalan, "On the Negative Pell Equation." International Journal of Mathematics Trends and Technology, vol. 21, no. 1, pp. 16-20, 2015.
[17] Manju Somanath, et al., "On a Class of Solutions for a Quadratic Diophantine Equation," Advances and Applications in Mathematical Sciences, vol. 19, no. 11, pp. 1097-1103, 2020.
[18] Sandhya P, and Pandichelvi V, "Assessment of Solutions in Pell and Pell-Lucas Numbers to Disparate Polynomial Equations of Degree Two," Wesleyan Journal of Research, vol. 14, no. 5, pp. 129-134, 2021.
[19] Aggarwal, Sudhanshu, and Sanjay Kumar, "On the Non-Linear Diophantine Equation 192m+(22r+1 -1) = ρ2," International Journal of Latest Technology in Engineering, Management & Applied Science, vol. 10, no. 2, pp. 14-16, 2021.
[20] Miller Steven J, and Ramin Takloo-Bighash, "An Invitation to Modern Number Theory," Princeton University Press, 2021.
[21] Refik Keskin, and Zafer Şiar. "A Note on Terai's Conjecture Concerning the Exponential Diophantine Equation x2+by=cz," arXiv preprint arXiv:2105.14814, 2021. Crossref, https://doi.org/10.48550/arXiv.2105.14814
[22] Vijayakumar P. N, and Sivaraman R, "On Solving Euler’s Quadratic Diophantine Equation," Journal of Algebraric Statistics, vol. 13, no. 3, pp. 815-817, 2022.
[23] Sengothai R, and R. Sivaraman, "Solving Diophantine Equations using Bronze Ratio," Journal of Algebraic Statistics, vol. 13, no. 3, pp. 812-814, 2022.
[24] Sivaraman R, "On Solving a Quadratic Diophantine Equation Involving Powers of 9," Bulletin of Mathematics and Statistics Research, vol. 10, no. 3, pp. 1-3, 2022. Crossref, https://doi.org/10.33329/bomsr.10.3.1
[25] Kannan J, et al., "Encryption Decryption Algorithm Using Solutions of Pell Equation," International Journal of Mathematics and its Applications, vol. 10, no. 1, pp. 1-8, 2022
V. Pandichelvi, S. Saranya, "Implication of Continued Fraction and Metallic Ratios to Resolve Binary Quadratic Diophantine Equations," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 12, pp. 13-15, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I12P503
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