Aigner has defined elite primes as primes p such that all but finitely many of Fermat numbers F(n) = 2²ⁿ + 1, n = 0,1,2,..., are quadratic nonresidues modulo p. Since the sequence of Fermat numbers is eventually periodic modulo any p with at most p distinct elements in the image, both the period length t_p and the number of arithmetic operations modulo p to test p for being elite are also bounded by p. We show that t_p = O(p^3/4), in particular improving the estimate t_p ≤ (p+1)/4 of Müller and Reinhart in 2008. The same bound O(p^3/4) also holds for testing anti-elite primes.