Fermat Primes

Fermat Primes
Basic Info
Discovered by	Pierre de Fermat
Number of	Exactly five- Fermat numbers apart from the first five are composite
Description	A prime which is one more than two to the power of a power of two
First Few	3, 5, 17, 257, 65,537

Fermat primes are a type of number that satisfies the properties of BOTH Fermat numbers and prime numbers. Just in case you haven't know yet, Fermat number has general form $F_n = 2^{2^n} + 1$.

All the Fermat primes

Note that 65,537 ($F_4$) is the largest known Fermat prime, being $2^{16} + 1$. However, The next Fermat number ($F_5 = 2^{32} + 1 = 4,294,967,297$) is divisible by 641 and 6,700,471.

Applications in Mathematics

Fermat primes are related to constructible polygons- regular polygons which one can construct using only a straightedge and circles. Polygons can be constructed if the number of sides' prime factors are either powers of two or powers of Fermat primes