I have the quadratic Diophantine equation: $$2x^2-y^2-y=0$$ $$x < y$$ and I'm writing a computer program which requires finding all positive integer solutions to this equation for $y\leq b$, where $b$ is a bound which could potentially be very large.
So far, the only way I seem to be able to solve this is by iterating over all $y\leq b$, solving for x and checking the result, which can be very slow.
Is there a more efficient way to do this? I've read that quadratic Diophantine equations can be represented as two Pell like equations which can be solved more easily but I have not been able to find a clear explanation of this.