# Fermat prime greater than three implies three is primitive root

## 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 .

## 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.