I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.

My question is: When does Mordell's Equation (i.e. y^2 = x^3 + k) have only FINITELY many solutions over the field of rational numbers, if we allow k itself to be a rational number? I've seen a criterion related to the class number of the (real) quadratic field Q(sqrt(k)), but it is limited only to k being either 1 or 2 modulo 4.