Negative Pell equation ($x^2-D{y}^2=-1$)

The Diophantine equation $x^2-D{y}^2=-1$, usually with a nonsquare positive integer $D$ and integer unknowns. It need not have a solution. For $D=2$, multiplying $u+v\sqrt{2}$ by $3+2\sqrt{2}$ preserves norm $-1$ and gives $(u,v)\mapsto (3u+4v,2u+3v)$.