Fermat prime
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
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.
Relation with other properties
Weaker properties
Complementary properties
Testing
The ID of the sequence in the Online Encyclopedia of Integer Sequences is A019434
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.