# Fermat prime implies every quadratic nonresidue is a primitive root

## Statement

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

## 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.