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A Fermat prime is a Fermat number
-
which happens to be prime.
is an
almost-square prime (i.e. primes of the form
) while Fermat primes
are a subset of the
quasi-square primes (i.e. primes of the form
). Thus all Fermat primes are
near-square prime (i.e. primes of the form
).
A019434 List of Fermat primes: primes of form
, for some
.
-
{3, 5, 17, 257, 65537, ?}
It is conjectured that there are only 5 terms. Currently it has been shown that
is
composite for
.
[1]
A092506 2 together with the Fermat primes, i.e. prime numbers of the form
2 n + 1, n = 0 or n = 2 k, k ≥ 0 |
.
-
{2, 3, 5, 17, 257, 65537, ?}
Product of distinct Fermat primes
An odd-sided
regular polygon is
constructible (with straightedge and compass) if and only if the number of sides is a
product of distinct Fermat primes. Since any angle can be bisected with straightedge and compass, an
-sided regular polygon
is thus constructible if and only if
is a
power of 2 (
) times a product of distinct Fermat primes (or empty product of Fermat primes).
A045544 Odd values of
for which a regular
-gon can be constructed by compass and straightedge.
-
{3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295, ?}
If there are no more Fermat primes, then 4294967295 is the last term in the sequence.
A003401 Numbers of edges of polygons constructible with ruler and compass.
-
{1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, ...}
The terms 1 and 2 correspond to degenerate polygons.
See also
Notes