The Diophantine equation $Dx^2+2^{2m+1}=y^{n}$

J. H. E. Cohn¹

¹ 23, Highfield Gardens London NW11 9HD, U.K.

Colloquium Mathematicum, Tome 98 (2003) no. 2, pp. 147-154.

Résumé

It is shown that for a given squarefree positive integer $D$, the equation of the title has no solutions in integers $x>0$, $m>0$, $n\ge 3$ and $y$ odd, nor unless $D\equiv 14 \ ({\rm mod}\hskip 1.7pt16)$ in integers $x>0$, $m=0$, $n\ge 3$, $y>0$, provided in each case that $n$ does not divide the class number of the imaginary quadratic field containing $\sqrt {-2D}$, except for a small number of (stated) exceptions.

Détail

DOI : 10.4064/cm98-2-1

Keywords: shown given squarefree positive integer equation title has solutions integers odd nor unless equiv mod hskip integers provided each does divide class number imaginary quadratic field containing sqrt except small number stated exceptions