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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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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
Email: bugeaud@pari.u-strasbg.fr

DOI: https://doi.org/10.1090/S0002-9939-97-04093-8
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