Fermat number

In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form

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

where n is a non-negative integer. The first few Fermat numbers are:

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... (sequence A000215 in the OEIS).

If 2^k + 1 is prime and k > 0, then k itself must be a power of 2, so 2^k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023, the only known Fermat primes are F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537 (sequence A019434 in the OEIS); heuristics suggest that there are no more.