On the equation \(x^ 2-Dy^ 2=z^ k\) with \(D=2,3,5,7,11,13\) (Q1118634)

The author proves that all integer solutions of the equation \(x^ 2- Dy^ 2=z^ k\), k is a positive integer, where \(D=2,3,5,7,11,13\), \((x,y,z)=1\), \(z\equiv 1 (mod 2)\) are given by \[ | z| =| m^ 2-Dn^ 2|,\quad x=((m+\sqrt{D}n)^ k(u+\sqrt{D}v)+(m-\sqrt{D}n)^ k(u-\sqrt{D}v)), \] \[ y=(1/2\sqrt{D})(m+\sqrt{D}n)^ k(u+\sqrt{D}v)-(m- \sqrt{D}n)^ k(u-\sqrt{D}v)), \] m,n,u,v\(\in {\mathbb{Z}}\), \((m,n)=1\) and \(| u^ 2-Dv^ 2| =1\) and in the case \(D=2,3,7,11\) the assumption that \(z\equiv 1 (mod 2)\) can be omitted.