Fermat number

In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:

F

n

=

2

2

n

+

1

,

{\displaystyle F_{n}=2^{2^{n}}+1,}

where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... (sequence A000215 in the OEIS).

If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. As of 2026, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in the OEIS).

Basic properties

The Fermat numbers satisfy the following recurrence relations:

F

n

=

(

F

n

−

1

−

1

)

2

+

1

{\displaystyle F_{n}=(F_{n-1}-1)^{2}+1}

F

n

=

F

0

F

1

⋯

F

n

−

1

+

2

{\displaystyle F_{n}=F_{0}F_{1}\cdots F_{n-1}+2}

for n ≥ 1,

F

n

=

F

n

−

1

+

2

2

n

−

1

F

0

F

1

⋯

F

n

−

2

{\displaystyle F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}F_{1}\cdots F_{n-2}}

F

n

=

(

F

n

−

1

−

2

F

n

−

2

+

1

)

2

+

(

2

(

F

n

−

2

−

2

)

(

F

n

−

3

−

1

)

)

2

+

(

2

F

n

−

2

−

3

)

2

{\displaystyle F_{n}=(F_{n-1}-2F_{n-2}+1)^{2}+(2(F_{n-2}-2)(F_{n-3}-1))^{2}+(2F_{n-2}-3)^{2}}

F

n

=

F

n

−

1

2

−

2

(

F

n

−

2

−

1

)

2

{\displaystyle F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}}

for n ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > 1. Then a divides both

F

0

F

1

⋯

F

j

−

1

{\displaystyle F_{0}F_{1}\cdots F_{j-1}}

and Fj; hence a divides their difference, 2.