Abstract
A prime is called elite, or anti-elite, when all but finitely many Fermat numbers are quadratic nonresidues or residues, respectively, modulo . It is known that if the multiplicative order of 2 modulo is of the form , where , then the prime is either elite or anti-elite. Modulo elite primes of this kind, we describe some criteria by which all sufficiently large Fermat numbers be primitive roots, or all nonprimitive roots.
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