We study prime numbers p that have the property that the numbers (b^q-1)/(b-1) are quadratic residues modulo p for all sufficiently large primes q. We give a practical criterion for testing for such primes with respect to the base b and, for certain values of b, we find a connection between these primes and anti-elite primes, i.e., where b²ⁿ+1 are quadratic residues modulo p for all large n.