Fermat primes

These are the primes of the form Fn = 2n+1
Not every value of n give primes, for instance 23+1 = 9 is of course not prime.
n should be a power of 2, but this is not sufficient : 232+1 = 4294967297 is a multiple of 641

Fn=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
2a p1p2p3...pn with the pi 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 :

2n+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 = 232 - 1 

strange, isn't it ? ...

 

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