# Fermat primes

These are the primes of the form F

_{n} = 2

^{n}+1

Not every value of n give primes, for instance
2

^{3}+1 = 9 is of course not prime.

n should be a power of 2, but this is not sufficient :
2

^{32}+1 = 4294967297 is a multiple of 641

F_{n}=3, 5, 17, 257, 65537, ...

Only these 5 Fermat primes are known, and it is unknown if there are others.

The Fermat primes are related to construction with compass and straightedge of a regular polygon with N sides.
To say more, a regular polygon with N sides can be constructed with compass and straightedge if and only if
the number of sides is

2^{a} p_{1}p_{2}p_{3}...p_{n}
with the p_{i} are different Fermat primes (a Gauss theorem).

Then polygons with 3,4,5,6,8,10,12,15,16,17,20... sides can be constructed

Polygons with 7,9,11,13,14,18,19... sides can't be constructed.

#### A strange relation :

2

^{n}+1 is prime for n = 1, 2, 4, 8, 16

the next power of 2 (32) doesn't result into a prime, as already mentionned. But...

3×5×17×257×65537 = 2^{32} - 1

strange, isn't it ? ...