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# Fermat numbers

### From OeisWiki

**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, ...}

## Contents |

## 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 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.)