# Fermat number

From Number

This article describes a sequence of natural numbers. The parameter for the sequence is a positive integer (or sometimes, nonnegative integer).

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This article defines a property that can be evaluated for a natural number, i.e., every natural number either satisfies the property or does not satisfy the property.

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## Contents

## Definition

Let be a nonnegative integer. The Fermat number, denoted , is defined as:

.

If it is prime, it is termed a Fermat prime.

## Occurrence

### Initial values

The initial values of are 3, 5, 17, 257, 65537, 4294967297 View list on OEIS

## Facts

- Composite Fermat number implies Poulet number: This states that any composite Fermat number is a Poulet number, i.e., a Fermat pseudoprime to base 2.
- Prime divisor of Fermat number is congruent to one modulo large power of two: For a Fermat number , any prime divisor is congruent to 1 mod , and for , congruen to 1 mod .
- Quadratic nonresidue equals primitive root for Fermat prime

## Relation with other properties

### Weaker properties

- Safe prime is a prime such that is also prime.
- Mersenne number is a number of the form , and a Mersenne prime is a Mersenne number that is also prime.