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Fermat numbers are numbers of the form
-
A000215 Fermat numbers:
.
-
{3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, ...}
A080176 Fermat numbers
in base
2 representation. (See
A080176, comment section.)
-
{11, 101, 10001, 100000001, 10000000000000001, 100000000000000000000000000000001, 10000000000000000000000000000000000000000000000000000000000000001, ...}
Formulae
-
Recurrences
-
F0 = 3; Fn = (Fn − 1 − 1) 2 + 1, n ≥ 1. |
-
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
.
Since there are infinitely many Fermat numbers, all mutually coprime, this implies that there are infinitely many prime numbers.
Generating function
-
Forward differences
-
Fn + 1 − Fn = (Fn ) 2 − 3 Fn + 2 = (Fn − 1) (Fn − 2), n ≥ 0. |
Partial sums
-
Fn = (2 2 n + 1) = m + 1 + 2 2 n = ?. |
Partial sums of reciprocals
-
Sum of reciprocals
-
Prime factorization of Fermat numbers
The prime factors of Fermat numbers are of the form[1]
-
Fn = 2 2 n + 1 = (k 1 ⋅ 2 n + 1) (k 2 ⋅ 2 n + 1) ⋯, k i ∈ ℕ +, n ≥ 0. |
Furthermore, the prime factors of Fermat numbers are of the form[2]
-
Fn = 2 2 n + 1 = ( ⋅ 2 n + 2 + 1) ( ⋅ 2 n + 2 + 1) ⋯, ∈ ℕ +, n ≥ 2. |
Prime factorization of Fermat numbers
|
|
Prime factors
|
0
|
3 = 2 1 + 1
|
3 = (1 / 2) ⋅ 2 2 + 1
|
1
|
5 = 2 2 + 1
|
5 = (1 / 2) ⋅ 2 3 + 1
|
2
|
17 = 2 4 + 1
|
17 = 1 ⋅ 2 4 + 1
|
3
|
257 = 2 8 + 1
|
257 = 8 ⋅ 2 5 + 1
|
4
|
65537 = 2 16 + 1
|
65537 = 1024 ⋅ 2 6 + 1
|
5
|
4294967297 = 2 32 + 1
|
641 ⋅ 6700417 = (5 ⋅ 2 7 + 1) (52347 ⋅ 2 7 + 1)
|
6
|
18446744073709551617 = 2 64 + 1
|
274177 ⋅ 67280421310721 = (1071 ⋅ 2 8 + 1) (262814145745 ⋅ 2 8 + 1)
|
7
|
340282366920938463463374607431768211457 = 2 128 + 1
|
59649589127497217 ⋅ 5704689200685129054721 = (116503103764643 ⋅ 2 9 + 1) (11141971095088142685 ⋅ 2 9 + 1)
|
F8 = 2 2 8 + 1 | = 115792089237316195423570985008687907853269984665640564039457584007913129639937 |
| = 1238926361552897 ⋅ 93461639715357977769163558199606896584051237541638188580280321 |
| = (1209889024954 ⋅ 2 10 + 1) (91271132534529275165198787304303609945362536661756043535430 ⋅ 2 10 + 1)
|
F9 = 2 2 9 + 1 | = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097 |
| = 2424833 ⋅ 7455602825647884208337395736200454918783366342657 ⋅ 741640062627530801524787141901937474059940781097519023905821316144415759504705008092818711693940737 |
| = (1184 ⋅ 2 11 + 1) (3640431067210880961102244011816628378312190597 ⋅ 2 11 + 1) (362128936829849024182024971631805407255830459520272960891514314523640507570656742232821636569307 ⋅ 2 11 + 1)
|
F10 = 2 2 10 + 1 | = 179769...137217 (309 decimal digits) |
| = 45592577 ⋅ 6487031809 ⋅ 4659775785220018543264560743076778192897 ⋅ P252* |
| = (11131 ⋅ 2 12 + 1) (1583748 ⋅ 2 12 + 1) (1137640572563481089664199400165229051 ⋅ 2 12 + 1) P252*
|
F11 = 2 2 11 + 1 | = 323170...230657 (617 decimal digits) |
| = 319489 ⋅ 974849 ⋅ 167988556341760475137 ⋅ 3560841906445833920513 ⋅ P564* |
| = (39 ⋅ 2 13 + 1) (119 ⋅ 2 13 + 1) (20506415569062558 ⋅ 2 13 + 1) (434673084282938711 ⋅ 2 13 + 1) P564*
|
* The subscripts show the number of decimal digits of the prime factor.
A093179 Smallest factor of the
-th Fermat number
.
-
{3, 5, 17, 257, 65537, 641, 274177, 59649589127497217, 1238926361552897, 2424833, 45592577, 319489, 114689, 2710954639361, 116928085873074369829035993834596371340386703423373313, 1214251009, ...}
A070592 Largest prime factor of the
-th Fermat number
.
-
{3, 5, 17, 257, 65537, 6700417, 67280421310721, 5704689200685129054721, 93461639715357977769163558199606896584051237541638188580280321, ...}
Fermat primes
It is conjectured that just the first 5 numbers in this sequence are primes (Fermat primes).
A019434 List of
Fermat primes:
primes of form
, for some
.
-
{3, 5, 17, 257, 65537, ?}
Products of distinct Fermat primes
Since there are
5 known Fermat primes,
{F0 , F1 , F2 , F3 , F4} = {3, 5, 17, 257, 65537} |
, then there are
() + () + () + () + () + () = 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2 5 |
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