where n is a natural
number. There are only five known Fermat primes: 3
(n=0), 5 (n=1),
17 (n=2),
257 (n=3) and 65537 (n=4). It is not known whether these are
the only Fermat primes, and it is not even known whether or not there are infinitely
many Fermat primes.
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Carl
Friedrich Gauss proved that there is a relationship between the ruler
and compass construction of regular polygons
and Fermat primes: a regular n-gon can be constructed with ruler and compasses
if and only if n is a power of 2 or the product of a power of 2 and distinct
Fermat primes.
with n a natural number are known as Fermat numbers. Fermat conjectured that all of them were prime numbers, but was later proven wrong when the Fermat number for n=5 was shown to be composite by Leonhard Euler in 1732. We have:
