Fermat primes

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A Fermat prime is a Fermat number

Fn  =  2 2 n + 1, n ≥ 0,

which happens to be prime.

F0 = 3

is an almost-square prime (i.e. primes of the form

k  2  −  1, k   ≥   1

) while Fermat primes

Fn , n   ≥   1,

are a subset of the quasi-square primes (i.e. primes of the form

k  2 + 1, k   ≥   1

). Thus all Fermat primes are near-square prime (i.e. primes of the form

k  2  ±  1, k   ≥   1

).
A019434 List of Fermat primes: primes of form

2 2 k + 1

, for some

k   ≥   0

.

{3, 5, 17, 257, 65537, ?}

It is conjectured that there are only 5 terms. Currently it has been shown that

2 2 k + 1

is composite for

5   ≤   k   ≤   32

.
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

n

-sided regular polygon

(n   ≥   3)

is thus constructible if and only if

n

is a power of 2 (

2 k, k   ≥   0

) times a product of distinct Fermat primes (or empty product of Fermat primes).
A045544 Odd values of

n

for which a regular

n

-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

• Mersenne primes
• Generalized Fermat primes
• Constructible polygons