# Quadratic nonresidue equals primitive root for Fermat prime

From Number

Template:Primitive root fact for special kind of prime

## Contents

## Statement

Let be a nonnegative integer such that the Fermat number is prime, i.e., is a Fermat prime. Then, if is a quadratic nonresidue modulo , then is a primitive root modulo .

## Related facts

- Prime divisor of Fermat number is congruent to one modulo large power of two
- Quadratic nonresidue that is not minus one is primitive root for safe prime
- Safe prime has plus or minus two as a primitive root

## Facts used

## Proof

**Given**: A Fermat prime , a quadratic nonresidue modulo .

**To prove**: is a primitive root modulo .

**Proof**: We need to show that the order of modulo is .

By fact (1), if is a quadratic nonresidue modulo , . Thus, the order of modulo does not divide . On the other hand, the order divides . Thus, the order of modulo equals , so is a primitive root modulo .