Fermat numbers

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Fermat numbers are numbers of the form

Fn  :=  2 2 n + 1, n ≥ 0.

A000215 Fermat numbers:

2 2 n + 1, n   ≥   0

.

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

A080176 Fermat numbers

2 2 n + 1, n   ≥   0,

in base 2 representation. (See A080176, comment section.)

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

Formulae

Fn  =  ?.

Recurrences

F0  =  3; Fn  =  (Fn  − 1 − 1)  2 + 1, n ≥ 1.

Fn  =    n  − 1 ∏   k  = 0    Fk + 2, n ≥ 0,

where for

n = 0

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

F0 = 3

, as expected.

Properties

The sequence of Fermat numbers is a coprime sequence, since

Fn  =    n  − 1 ∏   k  = 0    Fk + 2, n ≥ 0,

where for

n = 0

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

F0 = 3

.

Since there are infinitely many Fermat numbers, all mutually coprime, this implies that there are infinitely many prime numbers.

Generating function

G{Fn }(x)  =  ?.

Forward differences

Fn  + 1 − Fn  =  (Fn ) 2 − 3 Fn + 2  =  (Fn − 1) (Fn − 2), n ≥ 0.

Partial sums

m ∑   n   = 0    Fn  =    m ∑   n   = 0    (2 2 n + 1)  =  m + 1 +   m ∑   n   = 0    2 2 n  =  ?.

Partial sums of reciprocals

m ∑   n   = 0    1 Fn  =    m ∑   n   = 0    1 2 2 n + 1  =  ?.

Sum of reciprocals

∞ ∑   n   = 0    1 Fn  =    ∞ ∑   n   = 0    1 2 2 n + 1  =  ?.

Prime factorization of Fermat numbers

The prime factors of Fermat numbers are of the form

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

Fn  =  2 2 n + 1  =  (   k 1 4  ⋅  2 n  + 2 + 1)  (   k 2 4  ⋅  2 n  + 2 + 1) ⋯,  k i 4 ∈ ℕ +, n ≥ 2.

Prime factorization of Fermat numbers

n	Fn = 2 2 n + 1	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

n

-th Fermat number

Fn = 2 2 n + 1

.

{3, 5, 17, 257, 65537, 641, 274177, 59649589127497217, 1238926361552897, 2424833, 45592577, 319489, 114689, 2710954639361, 116928085873074369829035993834596371340386703423373313, 1214251009, ...}

A070592 Largest prime factor of the

n

-th Fermat number

Fn = 2 2 n + 1

.

{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

2 2 k + 1

, for some

k   ≥   0

.

{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

     

( 5 0 ) + ( 5 1 ) + ( 5 2 ) + ( 5 3 ) + ( 5 4 ) + ( 5 5 )  =  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

• Fermat primes
• Constructible odd-sided polygons
• Generalized Fermat numbers