# Sequence A000215, Fermat Numbers

The Fermat numbers, Sloane's A000215, are numbers
of the form 2^{2m}+1 for non-negative integer m.

The sequence begins: 2^{20}+1 = 3, 2^{21}+1 = 5, 2^{22}+1 =
17, 2^{23}+1 = 257, 2^{24}+1 = 65537, 2^{25}+1 = 4294967297,
2^{26}+1 = 18446744073709551617, 2^{27}+1 =
340282366920938463463374607431768211457, 2^{28}+1 =
115792089237316195423570985008687907853269984665640564039457584007913129639937,
2^{29}+1 =
13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097,
...

In 1640, Fermat knew that F_{0} through F_{4} are prime and
conjectured that all higher Fermat numbers were also prime. In the
years since the Fermat numbers and their factors have been the subject
of much research.

In 1732, Euler showed that all factors of a Fermat number F_{m} must
be of the form k×2^{m+1}+1. This limits the number of primes
that must be tested to factor a given Fermat number. It then becomes
easy to find that F_{5} = 641×6700417, because 641 is only the
5^{th} prime that satisfies the requirement of being equal to
k×2^{6}+1 for some k.

In 1801, Gauss proved that a regular polygon can be constructed by the
classical technique of straightedge and compass ("ruler and compass")
if and only if the number of sides is a product of 2^{n} and prime
Fermat numbers. Thus, for example, a 17-sided polygon can be
constructed, as can a 51-sided polygon (because 51=3×17) but a
7-sided or 21-sided polygon cannot because both are multiples of 7.

In 1878 Lucas improved on the Euler requirement by showing that a
Fermat factor must be of the form k×2^{m+2}+1. So for example,
the factors of F_{5} are of the form k×2^{7}+1 (and 641 is the
first prime that satisfies this requirement).

The factors of Fermat numbers are discussed here.

Some other sequences are discussed here.

Sources:

http://www.prothsearch.net/fermat.html

http://www.fermatsearch.org/history.htm

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2018 Aug 27. s.11