A GENERALISATION OF THE DIOPHANTINE EQUATION x^2+8∙7^b=y^2r

Article Sidebar

PDF
Published: Jun 30, 2021
DOI: https://doi.org/10.22452/mjs.vol40no2.3
Keywords:
Diophantine equation polynomial integral solution generator geometric sequence

Main Article Content

Siti Hasana Sapar
Universiti Putra Malaysia
Kai Siong Yow
Universiti Putra Malaysia

Abstract

We investigate the integral solutions to the Diophantine equation  where . We first generalise the forms of  and  that satisfy the equation. We then show the general forms of non-negative integral solutions to the equation under several conditions. We also investigate some special cases in which the integral solutions exist.

Downloads

Download data is not yet available.

Article Details

How to Cite
Sapar, S. H., & Yow, K. S. (2021). A GENERALISATION OF THE DIOPHANTINE EQUATION x^2+8∙7^b=y^2r. Malaysian Journal of Science, 40(2), 25–39. https://doi.org/10.22452/mjs.vol40no2.3
Issue
Vol 40 No 2 (2021): June
Section
Original Articles

Licensee MJS, Universiti Malaya, Malaysia. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

References

Abu Muriefah, F. S. & Bugeaud, Y. (2006). The Diophantine equation x^2+c=y^n: a brief overview. Revista Colombiana de Matemáticas. 40: 31-37.

Arif, S. A. and Abu Muriefah, F. S. (1998). The Diophantine equation x^2+3^m=y^n. International Journal of Mathematics and Mathematical Sciences, 21(3): 619-620.

Cohn, J. H. E. (1992). The Diophantine equation x^2+2^k=y^n. Arch. Math. (Basel). 59: 341-344.

Cohn, J. H. E. (1993). The Diophantine equation x^2+c=y^n. Acta Arithmetica, 65: 367-381.

Ko, C. (1965). On the Diophantine equation x^2=y^n+1,xy≠0. Scientia Sinica, 14: 457-460.

Le, M. (1997). Diophantine equation x^2+2^m=y^n. Chinese Science Bulletin, 42: 1515-1517.

Le, M. (2003). On the Diophantine equation x^2+p^2=y^n. Publ. Math. Debrecen. 63: 67- 78.

Lebesgue, V. A. (1850). Sur l’impossibilité en nombres entiers de l’équation x^m=y^2+1. Nouvella Annals Des Mathemetics, 78: 26-35.

Luca, F. (2000). On a Diophantine equation. Bulletin of the Australian Mathematical Society, 61: 241-246.

Luca, F. (2002). On the equation x^2+2^a 〖∙3〗^b=y^n. International Journal of Mathematics and Mathematical Sciences, 29: 239-244.

Luca, F. and Togbe, A. (2008). On the Diophantine equation x^2+2^a∙5^b=y^n. International Journal of Number Theory, 4: 973-979.

Mignotte, M. and Weger, B. M. M. (1996). On the Diophantine equation x^2+74=y^5 and x^2+86=y^5. Glasgow Mathematical Journal, 38(1): 77-85.

Wiles, A. J. (1995). Modular elliptic curves and Fermat’s Last Theorem. Annals of Mathematics, 141(3): 443-551.

Yow, K. S., Sapar, S. H. and Atan, K. A. (2013). On the Diophantine equation x^2+4∙7^b=y^2r. Pertanika J. Sci. & Technol, 21(2): 443-458.