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ON THE DIOPHANTINE EQUATION x2 + 5a 13b = yn

Published online by Cambridge University Press:  01 January 2008

FADWA S. ABU MURIEFAH,
FLORIAN LUCA and
ALAIN TOGBÉ
FADWA S. ABU MURIEFAH
Affiliation:
Mathematics Department, Riyadh University for Girls, P.O. Box 60561 Riyadh 11555Saudi Arabia e-mail: abumuriefah@yahoo.com
FLORIAN LUCA
Affiliation:
Instituto de Matemáticas UNAM, Campus Morelia Apartado Postal 27-3 (Xangari), C.P. 58089, Morelia, Michoacán, Mexico e-mail: fluca@matmor.unam.mx
ALAIN TOGBÉ
Affiliation:
Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville IN 46391USA e-mail: atogbe@pnc.edu
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Abstract

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In this note, we find all the solutions of the Diophantine equation x2 + 5a 13b = yn in positive integers x, y, a, b, n≥ 3 with x and y coprime.

Keywords

11D6111Y50

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 1 , January 2008, pp. 175 - 181
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Arif, S. A. and Abu Muriefah, F. S., On the Diophantine equation x 2 + 2k = y n, Internat. J. Math. Math. Sci. 20 (1997), 299–304.CrossRefGoogle Scholar
2.Arif, S. A. and Muriefah, F. S. Abu, The Diophantine equation x 2 + 3m = y n, Internat. J. Math. Math. Sci. 21 (1998), 619–620.CrossRefGoogle Scholar
3.Arief, S. A. and Muriefah, F. S. Abu, On a Diophantine equation, Bull. Austral. Math. Soc. 57 (1998), 189–198.Google Scholar
4.Abu Muriefah, F. S. and Arif, S. A., The Diophantine equation x 2 + 52k+1 = y n, Indian J. Pure Appl. Math. 30 (1999), 229–231.Google Scholar
5.Arif, S. A. and Muriefah, F. S. Abu, On the Diophantine equation x 2 + q 2k+1 = y n, J. Number Theory 95 (2002), 95–100.CrossRefGoogle Scholar
6.Abu Muriefah, F. S., On the diophantine equation x 2 +52k = y n, Demonstratio Mathematica 319 (2) (2006), 285–289.Google Scholar
7.Muriefah, F. S. Abu and Bugeaud, Y., The Diophantine equation x 2 + c = y n: a brief overview, Rev. Colombiana Math. 40 (2006), 31–37.Google Scholar
8.Bilu, Yu., Hanrot, G. and Voutier, P. M., Existence of primitive divisors of Lucas and Lehmer numbers, (Appendix by M. Mignotte), J. reine angew. Math. 539 (2001), 75–122.Google Scholar
9.Bugeaud, Y., Mignotte, M. and Siksek, S., Classical and modular approaches to exponential Diophantine equations. II. The Lebesgue-Nagell equation, Composition Math. 142 (2006), 31–62.CrossRefGoogle Scholar
10.Cohn, J. H. E., The Diophantine equation x 2 + c = y n, Acta Arith. 65 (1993), 367–381.CrossRefGoogle Scholar
11.Ko, C., On the Diophantine equation x 2 = y n +1, xy 0, Sci. Sinica 14 (1965), 457–460.Google Scholar
12.Le, M., An exponential Diophantine equation, Bull. Austral. Math. Soc. 64 (2001), 99–105.CrossRefGoogle Scholar
13.Le, M., On Cohn's conjecture concerning the Diophantine equation x 2 + 2m = y n, Arch. Math. (Basel) 78 (2002), 26–35.CrossRefGoogle Scholar
14.Lebesgue, V. A., Sur l'impossibilité en nombres entiers de l'équation x m = y 2+1, Nouv. Annal. des Math. 9 (1850), 178–181.Google Scholar
15.Luca, F., On a Diophantine Equation, Bull. Austral. Math. Soc. 61 (2000), 241–246.CrossRefGoogle Scholar
16.Luca, F., On the equation x 2 + 2a \cdot 3b = y n, Int. J. Math. Math. Sci. 29 (2002), 239–244.CrossRefGoogle Scholar
17.Luca, F. and Togbé, A. On the equation x 2 + 2a · 5b = y n, Int. J. Number Theory, to appear.Google Scholar
18.Mignotte, M. and Weger, B. M. M. de, On the Diophantine equations x 2+74 = y 5 and x 2+86=y5, Glasgow Math. J. 38 (1996), 77–85.CrossRefGoogle Scholar
19.Pink, I., On the diophantine equation x 2 + 2α. 3β. 5γ. 7δ =y n, Publ. Math. Debrecen 70/1–2 (2006), 149–166.Google Scholar
20.Tengely, Sz., On the Diophantine equation x 2 + a 2 = 2y p, Indag. Math. (N.S.) 15 (2004), 291–304.CrossRefGoogle Scholar