It's weird seeing Pell's equation discussed as something intractable or unsolved. We have deterministic ways of solving them (you can use convergents, which can be calculated using continuous fractions [0]). But then again, Diophantine equations sometimes show surprising patterns, and in this case the problem is not so much about solving the equation, but about finding cases where it doesn't have a solution. Which can be determined reasonably quickly for any given equation (at least, one with smallish coefficients), but apparently it's difficult to pinpoint an exact pattern of cases where this happens.

[0] https://en.wikipedia.org/wiki/Continued_fraction#Infinite_co...