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Fermat numbers∗
drini†
2013-03-21 12:19:17
The n-th Fermat number is defined as:
n
Fn = 22 + 1.
Fermat incorrectly conjectured that all these numbers were primes, although
he had no proof. The first 5 Fermat numbers: 3, 5, 17, 257, 65537 (corresponding
to n = 0, 1, 2, 3, 4) are all primes (so called Fermat primes) Euler was the first
to point out the falsity of Fermat’s conjecture by proving that 641 is a divisor
of F5 . (In fact, F5 = 641 × 6700417). Moreover, no other Fermat number
is known to be prime for n > 4, so now it is conjectured that those are all
prime Fermat numbers. It is also unknown whether there are infinitely many
composite Fermat numbers or not.
One of the famous achievements of Gauss was to prove that the regular
polygon of m sides can be constructed with ruler and compass if and only if m
can be written as
m = 2k Fr1 Fr2 · · · Frt
where k ≥ 0 and the other factors are distinct primes of the form Fn (of course,
t may be 0 here, i.e. m = 2k is allowed).
There are many interesting properties involving Fermat numbers. For instance:
Fm = F0 F1 · · · Fm−1 + 2
for any m ≥ 1, which implies that Fm − 2 is divisible by all smaller Fermat
numbers.
The previous formula holds because
m
m
m−1
Fm − 2 = (22 + 1) − 2 = 22 − 1 = (22
− 1)(22
m−1
m−1
+ 1) = (22
− 1)Fm−1
and expanding recursively the left factor in the last expression gives the desired
result.
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References.
Krı́zek, Luca, Somer. 17 Lectures on Fermat Numbers. CMS Books in Mathematics.
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