Fermatprime

Fermatprime is a term used to refer to a prime number that is also a Fermat number. A Fermat number is a number of the form $F_n = 2^{2^n} + 1$, where $n$ is a non-negative integer. The first few Fermat numbers are $F_0 = 2^{2^0} + 1 = 2^1 + 1 = 3$, $F_1 = 2^{2^1} + 1 = 2^2 + 1 = 5$, $F_2 = 2^{2^2} + 1 = 2^4 + 1 = 17$, $F_3 = 2^{2^3} + 1 = 2^8 + 1 = 257$, and $F_4 = 2^{2^4} + 1 = 2^{16} + 1 = 65537$.

It is known that $F_0$, $F_1$, $F_2$, $F_3$, and $F_4$ are all prime numbers. Therefore, 3, 5, 17, 257, and 65537 are Fermatprimes.

However, for $n \ge 5$, the Fermat numbers are not prime. This was first observed by Leonhard Euler, who showed that $F_5 = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297$ is divisible by 641. As of the current date, no other Fermat numbers $F_n$ for $n > 4$ have been found to be prime. It is an open question whether any Fermat numbers beyond $F_4$ are prime. If any are found to be prime, they would also be considered Fermatprimes. The study of Fermat numbers and their primality is a significant area in number theory, with implications for constructing regular polygons using ruler and compass.