A prime number p is called b-elite if only finitely many generalized Fermat numbers F_b,n = b²ⁿ+1 are quadratic residues to p. So far, only the case b = 2 was subjected to theoretical and experimental researches by several authors. Most of the results obtained for this special case can be generalized for all bases b > 2. Moreover, the generalization allows an insight to more general structures in which standard elite primes are embedded. We present selected computational results from which some conjectures are derived.