Fermat prime implies every quadratic nonresidue is a primitive root

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Statement

Suppose is a Fermat prime. Then, every quadratic nonresidue modulo is a primitive root modulo . The number of these elements is .

Related facts

Proof

Proof idea

Let .

The idea is that if an element is not a primitive root, then we must have , which, by Euler's criterion for quadratic residues, implies that is a quadratic residue. Thus, every quadratic nonresidue is a primitive root. The same argument establishes the converse.