Fermat number

Fermat prime
Named after	Pierre de Fermat
No. of known terms	5
Conjectured no. of terms	5
Subsequence of	Fermat numbers
First terms	3, 5, 17, 257, 65537
Largest known term	65537
OEIS index	A019434

In mathematics, a Fermat number, named after Pierre de Fermat (1607–1665), the first known to have studied them, is a positive integer of the form: $F_{n}=2^{2^{n}}+1,$ where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... (sequence A000215 in the OEIS).

If 2^k + 1 is prime and k > 0, then k itself must be a power of 2,^[1] so 2^k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023^[update], the only known Fermat primes are F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537 (sequence A019434 in the OEIS).

^ For any positive odd number $m$ , $2^{2^{k}m}+1=(a+1)(a^{m-1}-a^{m-2}+\ldots -a+1)$ where $a=2^{2^{k}}$ .

[1] For any positive odd number $m$ , $2^{2^{k}m}+1=(a+1)(a^{m-1}-a^{m-2}+\ldots -a+1)$ where $a=2^{2^{k}}$ .

[1]