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

Volume 98 / 2003

J. H. E. Cohn Colloquium Mathematicum 98 (2003), 147-154 MSC: Primary 11D61; Secondary 11B37. DOI: 10.4064/cm98-2-1

Abstract

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.

Authors

  • J. H. E. Cohn23, Highfield Gardens
    London NW11 9HD, U.K.
    e-mail

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