Fermat number

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

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.
View a complete list of properties of natural numbers

Definition

Let $n$ be a nonnegative integer. The $n^{th}$ Fermat number, denoted $F_{n}$, is defined as:

$F_{n}:=2^{2^{n}}+1$.

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

Occurrence

Initial values

The initial values are $F_{0}=3,F_{1}=5,F_{2}=17,F_{3}=257,F_{4}=65537$.

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A000215

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 $F_{k}=2^{2^{k}}+1$, any prime divisor is congruent to 1 mod $2^{k+1}$, and for $k\geq 2$, congruen to 1 mod $2^{k+2}$.
• Quadratic nonresidue equals primitive root for Fermat prime

Relation with other properties

Weaker properties

• Proth number
• Generalized Fermat number

Other related properties

• Safe prime is a prime $p$ such that $(p-1)/2$ is also prime.
• Mersenne number is a number of the form $2^{n}-1$, and a Mersenne prime is a Mersenne number that is also prime.