Given $a,b,c>0$, is there a procedure to solve $(x,y)\in\Bbb Z:ax^2+by^2=c$ in $O(\log^d c)$ arithmetic operations (either randomized or deterministic) with $d>0$ being fixed?
• looking in Dickson's History, it seems likely that this refers to an indefinite form, that is your $ab < 0.$ The method is the one i have answered with, over and over, on MSE for Pell and similar equations. Meanwhile, I think the Disquisitiones has been translated, the two mentions in the History are Article 195 and Article 323, where the phrase "method of exclusions" goes with 323. Note that the positive form case and indefinite are completely different in terms of solving such problems. – Will Jagy Apr 1 '15 at 17:08