# Fermat prime greater than three implies three is primitive root

From Number

## Contents

## Statement

Suppose is a *positive* integer (in particular, it is not zero) such that the Fermat number is a prime number (and hence a Fermat prime). Then, the number 3 is a quadratic nonresidue modulo , and hence a primitive root modulo .

## Related facts

- Quadratic nonresidue equals primitive root for Fermat prime
- Safe prime has plus or minus two as a primitive root

## Facts used

## Proof

We use Fact (1), which says that, in terms of Legendre symbols:

Note that:

- Since , , so the exponent on the right side is even, so the right side is 1.
- We know that , so , so . Thus, which is -1.

Plugging these in, we get:

Thus, 3 is a quadratic nonresidue mod . Combine with Fact (2) to obtain that it is a primitive root.