Quadratic nonresidue equals primitive root for Fermat prime

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
• Safe prime has plus or minus two as a primitive root

Facts used

1. Euler's criterion

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 .