Let S = {a1, a2, . . . , an} be a set of nonzero integers such that for any nonempty subset T of S, the product of all the elements in T is not a perfect square. Then the density of the set of primes p for which the ai’s are quadratic non-residues modulo p, but not primitive roots modulo p, is at least 1 2n(q 1)qm , where m is a non-negative integer with m ? n and q is the least odd prime which does not divide ai for all i = 1, 2, . . . , n.
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