On the Diophantine Equation X<sup>2</sup> + 13<sup>K</sup> = Y<sup>N</sup>

Authors

  • Abdelkader Hamtat A.T.N. Laboratory, USTHB University Algiers
  • Djilali Behloul A.T.N. Laboratory, USTHB University Algiers

DOI:

https://doi.org/10.18311/jims/2017/15570

Keywords:

Primitive Divisor Theorem of Carmicheal, Theorem of Catalan, Nagell Equation

Abstract

The object of this paper is to find all solutions of the dio-phantine equation x2 + 13k = yn, in positive integers x, y with n ≥ 3.

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References

F.S. Abu Muriefah, F. Luca, A. Togbe, On the Diophantine equation x2 + 5a13b = yn, Glasgow Math. J. 50 (2008), 175-181.

S.A. Arif and F. S. Abu Muriefah, On the Diophantine equation x2 + q2k+1 = yn, The Arabian J. for Sci. and Engineering, 26(1A), 53-62, 2001.

S.A. Arif and F.S. Abu Muriefah, On the Diophantine equation x2 + q2k = yn, Journal of Number Theory, 95(1), 95-100, 2002.

A. Berczes, I. Pink, On the Diophantine equation x2 +p2k = yn, Arch. Math. 91 (2008) 505-517.

Y.F. Bilu, G. Hanrot, P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539, 75-122, 2001.

Y. Bugeaud, F. Luca, M. Mignotte, S. Siksek, Almost Powers in the Lucas Sequence. Journal de Theorie des Nombres de Bordeaux. 20 (2008), no 3, 555-600.

H. Cohen, Explicit Methods for Solving Diophantine Equations. Tucson, Arizona Winter School, 2006.

J.H.E. Cohn, The Diophantine equation x2 + C = yn, Acta Arith. 65 (1993), No.4, 367-381.

J.H.E. Cohn, The Diophantine equation x2+C = yn, Acta Arith., 65(4), 367-381, 1995.

E. Goins, F. Luca, A. Togbe, On the Diophantine equation x2 + 2α5β13γ = yn, ANTS VIII Proceedings: A. J. van der Poorten and A. Stein (eds.), ANTS VIII, Lecture Notes in Computer Science 5011 (2008), 430-442.

Hui Lin Zhu, Mao Hua Lec, On some generalized Lebesgue-Nagell equations. Journal of Number Theory 131 (2011) 458-469.

F. Luca, On the Equation x2 +2a3b = yn, Int. J. Math. Math. Sci. 29.4 (2002) 239-244

F. Luca, A. Togbe, On the Diophantine equation x2 +72k = yn, Fibonacci Quart., no. 4, 322-326 (2008).

F. Luca, Eective methods for Diophantine equations, Universidad Nacional Autonoma de Mexico (2009)

T. Nagell, Contributions to the theory of a category of Diophantine equations of the second degree with two unknown, Nova Acta Reg. Soc. Upsal. Ser. 4 (1955), no.16, 1-38.

S. Siksek, J.E. Cremona, On the Diophantine equation x2 + 7 = ym, Acta Arith. 109 (2) (2003) 143-149.

L. Tao, On the Diophantine equation x2 + 5m = yn, Ramanujan J. 19 (2009) 325-338.

Published

2017-07-01

How to Cite

Hamtat, A., & Behloul, D. (2017). On the Diophantine Equation X<sup>2</sup> + 13<sup>K</sup> = Y<sup>N</sup>. The Journal of the Indian Mathematical Society, 84(3-4), 191–200. https://doi.org/10.18311/jims/2017/15570