# Difference between revisions of "Fermat prime"

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==Definition== | ==Definition== | ||

− | A '''Fermat prime''' is a [[Fermat number]] that is also a [[prime number]]. In other words, it is a prime number of the form <math>F_k = 2^{2^k} + 1</math>, where <math>k</math> is a nonnegative integer. | + | A '''Fermat prime''' is a [[defining ingredient::Fermat number]] that is also a [[prime number]]. In other words, it is a prime number of the form <math>F_k = 2^{2^k} + 1</math>, where <math>k</math> is a nonnegative integer. |

It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>. | It turns out that if <math>2^n + 1</math> is prime for <math>n</math> a natural number, then <math>n = 2^k</math> for some nonnegative integer <math>k</math>. Thus, a Fermat prime can also be defined as a prime of the form <math>2^n + 1</math> for some natural number <math>n</math>. | ||

+ | |||

+ | ==Facts== | ||

+ | |||

+ | * [[Quadratic nonresidue equals primitive root for Fermat prime]] | ||

+ | * [[Fermat prime greater than three implies three is primitive root]] (we prove this by first showing that 3 is a quadratic nonresidue, then using the above). | ||

+ | * [[Proth's theorem]] | ||

+ | * [[Pepin's primality test]] | ||

+ | |||

+ | ===Related facts about (not necessarily prime) Fermat numbers=== | ||

+ | |||

+ | * [[Goldbach's theorem]]: The Fermat numbers are pairwise relatively prime. | ||

+ | * [[Prime divisor of Fermat number is congruent to one modulo large power of two]] | ||

+ | |||

+ | ==Occurrence== | ||

+ | |||

+ | ===Finitude conjecture=== | ||

+ | |||

+ | {{further|[[Finitude conjecture for Fermat primes]]}} | ||

+ | |||

+ | The only known Fermat primes are the Fermat primes for <math>k = 0,1,2,3,4</math>, namely, the primes <math>3,5,17,65537</math>. For all <math>k \ge 5</math>, either the [[Fermat prime]] <math>F_k</math> is known to be composite or its primality is open. The [[prime number theorem]] suggests that there are likely to be only finitely many Fermat primes, and it is conjectured that there are no Fermat primes other than the five known ones. | ||

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+ | {{oeis|A019434}} | ||

+ | |||

+ | ==Relation with other properties== | ||

+ | |||

+ | ===Weaker properties=== | ||

+ | |||

+ | * [[Stronger than::Proth prime]] | ||

+ | |||

+ | ===Complementary properties=== | ||

+ | |||

+ | * [[Composite Fermat number]] | ||

+ | ==Testing== | ||

+ | |||

+ | |||

+ | |||

+ | {{test|Pepin's test}} | ||

+ | |||

+ | Pepin's test is a deterministic primality test for Fermat numbers. It relies on [[Proth's theorem]], and the observation that <math>3</math> is a [[quadratic nonresidue]] for any Fermat prime greater than <math>3</math> itself. |

## Latest revision as of 19:07, 3 January 2012

This article defines a property that can be evaluated for a prime number. In other words, every prime number either satisfies this property or does not satisfy this property.

View other properties of prime numbers | View other properties of natural numbers

## Contents

## Definition

A **Fermat prime** is a Fermat number that is also a prime number. In other words, it is a prime number of the form , where is a nonnegative integer.

It turns out that if is prime for a natural number, then for some nonnegative integer . Thus, a Fermat prime can also be defined as a prime of the form for some natural number .

## Facts

- Quadratic nonresidue equals primitive root for Fermat prime
- Fermat prime greater than three implies three is primitive root (we prove this by first showing that 3 is a quadratic nonresidue, then using the above).
- Proth's theorem
- Pepin's primality test

### Related facts about (not necessarily prime) Fermat numbers

- Goldbach's theorem: The Fermat numbers are pairwise relatively prime.
- Prime divisor of Fermat number is congruent to one modulo large power of two

## Occurrence

### Finitude conjecture

`Further information: Finitude conjecture for Fermat primes`

The only known Fermat primes are the Fermat primes for , namely, the primes . For all , either the **Fermat prime** is known to be composite or its primality is open. The prime number theorem suggests that there are likely to be only finitely many Fermat primes, and it is conjectured that there are no Fermat primes other than the five known ones.

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A019434

## Relation with other properties

### Weaker properties

### Complementary properties

## Testing

### Test for this property

The test used to determine whether a natural number has this property is:Pepin's test

Pepin's test is a deterministic primality test for Fermat numbers. It relies on Proth's theorem, and the observation that is a quadratic nonresidue for any Fermat prime greater than itself.