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On the diophantine equation $x^{2} - 2^{m} = \pm \, y^{n}$

Author: Yann Bugeaud
Journal: Proc. Amer. Math. Soc. 125 (1997), 3203-3208
MSC (1991): Primary 11D61, 11J86
MathSciNet review: 1422850
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Abstract: One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation $x^{2} - 2^{m} = y^{n}$. Moreover, we prove that the diophantine equation $x^{2} - 2^{m} = \pm \, y^{n}$, $x$, $y$, $m$, $n \in \mathbf {N}$, gcd$(x, y) =1$, $y>1$, $n>2$ has only finitely many solutions, all of which satisfying $n \le 7.3 \, 10^{5}$.

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  • [1] Y. Bugeaud and M. Laurent, Minoration effective de la distance $p$-adique entre puissances de nombres algébriques, J. Number Th. 61 (1996), 311-342.
  • [2] Yongdong Guo and Maohua Le, A note on the exponential diophantine equation $x^{2} - 2^{m} = y^{n}$, Proc. Amer. Math. Soc. 123 (1995), 3627-3629. MR 96b:11040
  • [3] S. Hyyrö, Ueber das Catalansche Problem, Ann. Univ. Turku, Ser AI, 79 (1964), 3-10. MR 31:3378
  • [4] S. V. Kotov, Ueber die maximale Norm der Idealteiler des polynoms $\alpha x^{m} + \beta y^{n}$ mit den algebraischen Koeffizienten, Acta Arith. 31 (1976), 219-230. MR 55:261
  • [5] M. Laurent, M. Mignotte and Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Th. 55 (1995), 285-321. MR 96h:11073
  • [6] Maohua Le, The Diophantine Equation $x^{2} + D^{m} = 2^{n+ 2}$, Comment. Univ. St Pauli 43 (1994), 127-133.
  • [7] T. N. Shorey, A. J. Van der Poorten, R. Tijdeman and A. Schinzel, Applications of the Gel'fond-Baker method to diophantine equations, in Transcendence Theory : Advances and Applications, Academic Press, London (1977), 59-78. MR 57:12383

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Additional Information

Yann Bugeaud
Affiliation: Université Louis Pasteur, U. F. R. de mathématiques, 7, rue René Descartes, 67084 Strasbourg, France
Address at time of publication: 31 rue de l’Etang, 56600 Lanester, France

Keywords: Exponential equations, linear forms in logarithms
Received by editor(s): June 13, 1996
Communicated by: William W. Adams
Article copyright: © Copyright 1997 American Mathematical Society