Quadratic nonresidue equals primitive root for Fermat prime

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Template:Primitive root fact for special kind of 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

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 .