We can indeed relate this problem to the “usual” Diophantine approximation problem of approximating a number α by a rational number p/q, but there's a twist. T

We can indeed relate this problem to the “usual” Diophantine approximation problem of approximating a number α by a rational number p/q, but there's a twist. The “usual” problem, where we want p/q to be close to a real number α, can be thought of as finding Gaussian integers (q + ip) that lie close to the line y=αx in the complex plane, i.e. points z on the lattice such that Im(z)/Re(z) ≈ α. Here, with α=√3, we want to solve the same problem, i.e. find rational numbers p/q close to α, except that p/q must further be of the form (ad-bc)/(ac+bd) for some (a, b, c, d). Here's the thing: this is precisely the same as saying that (q + ip) is not a Gaussian prime! This is because (a-ib)(c+id) = (ac+bd) + i(ad-bc). (https://en.wikipedia.org/w/index.php?title=Brahmagupta%E2%80%93Fibonacci_identity&oldid=863796842 https://en.wikipedia.org/w/index.php?title=Brahmagupta%E2%80...) So any (q + ip) of the form (ac+bd) + i(ad-bc) can be written as the product of two Gaussian integers, and vice-versa. So the relation between the Diophantine approximation problem and this one is that while there we want to find Gaussian integers close to the line Im(z)=αRe(z), here we want to find composite Gaussian integers close to the line Im(z)=αRe(z). (Note we can get different "best" approximations depending on how we measure closeness; see my other comment here: https://news.ycombinator.com/item?id=18555240 https://news.ycombinator.com/item?id=18555240 Also, I expect that as denominators get larger, the primes get sparser, so this becomes closer to the usual problem. Of course in the original post linked here, the goal is to find small solutions, so the difference matters.)