[svat]

Right, this is about the negative Pell's equation, x^2 - dy^2 = -1 (rather than +1). The question is for which d this has a solution. (For the usual +1 case, which is what you're talking about, there are solutions for every d, and indeed continued fractions can be used.)

There are no solutions when d has any factor that is 3 mod 4, and the result mentioned in the article (Stevenhagen's conjecture, now proved by Koymans and Pagano) is that there are solutions for about 58% of the rest, specifically 1 - product_{j odd}(1 - 1/2^j) ≈ 0.580577558…. The first couple of pages of the paper (linked from the article) are actually very readable without much background: https://arxiv.org/pdf/2201.13424.pdf