Does anyone knows whether Brahmagupta's composition is applicable here or not? I'm stuck with not being able to generate 85 120 from 3 4 and 15 21.

Even completing squares to arrive at Pell's equation with integer is troublesome with like 3 8. No clue yet.

EDIT: It gets frustrating. This time I follow the continued fraction approach. The trouble is still at the very beginning. While now I know that as a perfect square should result in No solution, I could not get the Pell's Equation in its appropriate form. The one I've got is , where . So now it is . But when I plug in with and for the case and , those and are not integers. The fact is that, I couldn't even get the fundamental solutions for and right.

Is it possible that there are multiple valid Pell's Equations for one input? But then there would be infinitely many such equations as I could just modify the corresponding . This idea sounds horrifying. And when sqrt is involved, answers like 181489708628619 296371453137673 is found for 3 8 1000, which is obviously wrong.

Also, there is the issue with OverflowError: long int too large to convert to float for Test Case #11, #16 and #17, when I tried to perform y * sqrt(k) + 1.

Trying to implement something like the Generic two integer variable equation solver is too overwhelming.

EDIT 2: Just found this, this, this, and this. That's a lot of reading ahead of me.