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
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 .
- Quadratic nonresidue equals primitive root for Fermat prime
- Fermat prime 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
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
Test for this property
The test used to determine whether a natural number has this property is: Pepin's test