On the diophantine equation 𝑥²-2^{𝑚}=±𝑦ⁿ

Bugeaud, Yann

doi:10.1090/S0002-9939-97-04093-8

On the diophantine equation $x^2-2^m=\pm y^n$
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by Yann Bugeaud

Proc. Amer. Math. Soc. 125 (1997), 3203-3208

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

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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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