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Fermat numbers

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Fermat numbers are numbers of the form

Fermat numbers: (Cf. A000215)

{3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, ...}

Fermat numbers in base 2 representation. (Cf. A080176 comment)

{11, 101, 10001, 100000001, 10000000000000001, 100000000000000000000000000000001, 10000000000000000000000000000000000000000000000000000000000000001, ...}

Formulae

Recurrences

where for we have the empty product (giving the multiplicative identity, i.e. 1) + 2, giving , as expected.

Properties

The sequence of Fermat numbers is a coprime sequence, since

where for we have the empty product (giving the multiplicative identity, i.e. 1) + 2, giving

Generating function

Forward differences

Partial sums

Partial sums of reciprocals

Sum of reciprocals

Prime factorization of Fermat numbers

The prime factors of Fermat numbers are of the form (since they are odd)[1]

More interestingly, the prime factors of Fermat numbers are of the form

Prime factorization of Fermat numbers
Prime factors
0 3 = 1*2^1+1 3 = 1*2^1+1 = (1*(2*1)+1)
1 5 = 1*2^2+1 5 = 1*2^2+1 = (1*(2*2)+1)
2 17 = 1*2^4+1 17 = 1*2^4+1 = (2*(2*4)+1)
3 257 = 1*2^8+1 257 = 1*2^8+1 = (16*(2*8)+1)
4 65537 = 1*2^16+1 65537 = 1*2^16+1 = (2048*(2*16)+1)
5 4294967297 = 1*2^32+1 641 * 6700417 = (5*2^7+1) (52347*2^7+1) = (10*(2*32)+1) (104694*(2*32)+1)
6 18446744073709551617 = 1*2^64+1 274177 * 67280421310721 = (1071*2^8+1) (262814145745*2^8+1) = (2142*(2*64)+1) (525628291490*(2*64)+1)
7 340282366920938463463374607431768211457 = 1*2^128+1 59649589127497217 * 5704689200685129054721 = (116503103764643*2^9+1) (11141971095088142685*2^9+1) = (233006207529286*(2*128)+1) (22283942190176285370*(2*128)+1)


Fermat primes

It is conjectured that just the first 5 numbers in this sequence are primes (Fermat primes.)

List of Fermat primes: primes of form , for some . (Cf. A019434)

{3, 5, 17, 257, 65537, ?}

Products of distinct Fermat primes

Since there are 5 known Fermat primes, {, , , , } = {3, 5, 17, 257, 65537}, then there are

products of distinct known Fermat primes. The 31 non-empty products of distinct known Fermat primes give the number of sides of constructible odd-sided polygons (since a polygon has at least 3 sides.)

See also

Notes