It is shown that for a given squarefree positive integer D, the equation of the title has no solutions in integers x > 0, m > 0, n ≥ 3 and y odd, nor unless D ≡ 14 (mod 16) in integers x > 0, m = 0, n ≥ 3, y > 0, provided in each case that n does not divide the class number of the imaginary quadratic field containing √(-2D), except for a small number of (stated) exceptions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-1,
author = {J. H. E. Cohn},
title = {The Diophantine equation $Dx2 + 2^{2m+1} = yn$
},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {147-154},
zbl = {1053.11030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-1}
}
J. H. E. Cohn. The Diophantine equation $Dx² + 2^{2m+1} = yⁿ$
. Colloquium Mathematicae, Tome 96 (2003) pp. 147-154. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-1/