# Difference between revisions of "Quadratic nonresidue equals primitive root for Fermat prime"

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==Facts used== | ==Facts used== |

## Revision as of 14:43, 21 April 2009

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

## 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 .