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| In [[mathematics]], a '''Fermat number''', named after [[Pierre de Fermat]] who first studied them, is a [[natural number|positive integer]] of the form
| | I'm Rudolph (22) from San Jose, United States. <br>I'm learning Norwegian literature at a local high school and I'm just about to graduate.<br>I have a part time job in a the office.<br><br>Also visit my web site: [http://2mingle.in/blog/view/89066/how-to-get-free-fifa-15-coins Fifa 15 Coin Generator] |
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| :<math>F_{n} = 2^{(2^n)} + 1</math>
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| where ''n'' is a [[nonnegative]] integer. The first few Fermat numbers are:
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| : 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … {{OEIS|id=A000215}}.
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| If 2<sup>''n''</sup> + 1 is [[prime number|prime]], and ''n'' > 0, it can be shown that ''n'' must be a power of two. (If ''n'' = ''ab'' where 1 ≤ ''a'', ''b'' ≤ ''n'' and ''b'' is odd, then 2<sup>''n''</sup> + 1 = (2<sup>''a''</sup>)<sup>''b''</sup> + 1 ≡ (−1)<sup>''b''</sup> + 1 = 0 ('''mod''' 2<sup>''a''</sup> + 1). See [[#Other theorems about Fermat numbers|Sec. 5]] for complete proof.) In other words, every prime of the form 2<sup>''n''</sup> + 1 is a Fermat number, and such primes are called '''Fermat primes'''. The only known Fermat primes are ''F''<sub>0</sub>, ''F''<sub>1</sub>, ''F''<sub>2</sub>, ''F''<sub>3</sub>, and ''F''<sub>4</sub> {{OEIS|id=A019434}}.
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| ==Basic properties==
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| The Fermat numbers satisfy the following [[recurrence relation]]s:
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| :<math>
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| F_{n} = (F_{n-1}-1)^{2}+1\!</math>
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| for ''n'' ≥ 1,
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| :<math>
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| F_{n} = F_{n-1} + 2^{2^{n-1}}F_{0} \cdots F_{n-2}\!</math>
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| :<math>
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| F_{n} = F_{n-1}^2 - 2(F_{n-2}-1)^2\!</math>
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| :<math>
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| F_{n} = F_{0} \cdots F_{n-1} + 2\!</math>
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| for ''n'' ≥ 2. Each of these relations can be proved by [[mathematical induction]]. From the last equation, we can deduce '''Goldbach's theorem''' (named after [[Christian Goldbach]]): no two Fermat numbers [[coprime|share a common factor]]. To see this, suppose that 0 ≤ ''i'' < ''j'' and ''F''<sub>''i''</sub> and ''F''<sub>''j''</sub> have a common factor ''a'' > 1. Then ''a'' divides both
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| :<math>F_{0} \cdots F_{j-1}</math>
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| and ''F''<sub>''j''</sub>; hence ''a'' divides their difference, 2. Since ''a'' > 1, this forces ''a'' = 2. This is a [[contradiction]], because each Fermat number is clearly odd. As a [[corollary]], we obtain another proof of the [[infinity|infinitude]] of the prime numbers: for each ''F''<sub>''n''</sub>, choose a prime factor ''p''<sub>''n''</sub>; then the sequence {''p''<sub>''n''</sub>} is an infinite sequence of distinct primes.
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| Further properties:
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| *The number of digits ''D''(''n'',''b'') of ''F''<sub>''n''</sub> expressed in the [[numeral system|base]] ''b'' is
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| :<math>D(n,b) = \left\lfloor \log_{b}\left(2^{2^{\overset{n}{}}}+1\right)+1 \right\rfloor \approx \lfloor 2^{n}\,\log_{b}2+1 \rfloor </math> (See [[floor function]]).
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| *No Fermat number can be expressed as the sum of two [[prime number|primes]], with the exception of F<sub>1</sub> = 2 + 3.
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| *No Fermat prime can be expressed as the difference of two ''p''th powers, where ''p'' is an odd prime.
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| *With the exception of F<sub>0</sub> and F<sub>1</sub>, the last digit of a Fermat number is 7.
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| * The sum of the reciprocals of all the Fermat numbers {{OEIS|id=A051158}} is [[irrational number|irrational]]. ([[Solomon W. Golomb]], 1963)
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| ==Primality of Fermat numbers==
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| Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who [[conjecture]]d (but admitted he could not prove) that all Fermat numbers are prime. Indeed, the first five Fermat numbers ''F''<sub>0</sub>,...,''F''<sub>4</sub> are easily shown to be prime. However, this conjecture was refuted by [[Leonhard Euler]] in 1732 when he showed that
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| :<math> F_{5} = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297 = 641 \times 6700417. \; </math>
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| Euler proved that every factor of ''F''<sub>''n''</sub> must have the form ''k''2<sup>''n''+1</sup> + 1 (later improved to ''k''2<sup>''n''+2</sup> + 1 by [[Édouard Lucas|Lucas]]).
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| The fact that 641 is a factor of ''F''<sub>5</sub> can be easily deduced from the equalities 641 = 2<sup>7</sup>×5+1 and 641 = 2<sup>4</sup> + 5<sup>4</sup>. It follows from the first equality that 2<sup>7</sup>×5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 2<sup>28</sup>×5<sup>4</sup> ≡ 1 (mod 641). On the other hand, the second equality implies that 5<sup>4</sup> ≡ −2<sup>4</sup> (mod 641). These [[Modular arithmetic|congruences]] imply that −2<sup>32</sup> ≡ 1 (mod 641).
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| It is widely believed that Fermat was aware of the form of the factors later proved by Euler, so it seems curious why he failed to follow through on the straightforward calculation to find the factor.<ref>{{Harvnb|Křížek|Luca|Somer|2001|p=38, Remark 4.15}}</ref> One common explanation is that Fermat made a computational mistake and was so convinced of the correctness of his claim that he failed to double-check his work.
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| There are no other known Fermat primes ''F''<sub>''n''</sub> with ''n'' > 4. However, little is known about Fermat numbers with large ''n''.<ref>Chris Caldwell, [http://primes.utm.edu/links/theory/special_forms/ "Prime Links++: special forms"] at The [[Prime Pages]].</ref> In fact, each of the following is an open problem:
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| *Is ''F''<sub>''n''</sub> [[composite number|composite]] for all ''n'' > 4?
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| *Are there infinitely many Fermat primes? ([[Gotthold Eisenstein|Eisenstein]] 1844)<ref>{{Harvnb|Ribenboim|1996|p=88}}.</ref>
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| *Are there infinitely many composite Fermat numbers?
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| {{As of|2010}} it is known that ''F''<sub>''n''</sub> is composite for 5 ≤ ''n'' ≤ 32, although complete factorizations of ''F''<sub>''n''</sub> are known only for 0 ≤ ''n'' ≤ 11, and there are no known factors for ''n'' = 20 and ''n'' = 24.<ref name="Keller"/> The largest Fermat number known to be composite is ''F''<sub>2747497</sub>, and its prime factor 57×2<sup>2747499</sup> + 1 was discovered by Marshall Bishop in [[PrimeGrid]]'s Proth Prime Search on May 13, 2013.<ref name="PrimeGrid’s
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| Proth Prime Search - 57*2^2747499+1">{{cite web|title=PrimeGrid’s Proth Prime Search - 57*2^2747499+1 (official announcement)|url=http://www.primegrid.com/download/PPS-2747497.pdf|publisher=Primegrid|accessdate=26 May 2013}}</ref>
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| ===Heuristic arguments for density===
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| The following [[heuristic argument]] suggests there are only finitely many Fermat primes: according to the [[prime number theorem]], the "[[probability]]" that a number ''n'' is prime is at most ''A''/ln(''n''), where ''A'' is a fixed [[Constant (mathematics)|constant]]. Therefore, the total [[expected value|expected number]] of Fermat primes is at most
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| : <math>\begin{align}A \sum_{n=0}^{\infty} \frac{1}{\ln F_{n}} &= \frac{A}{\ln 2} \sum_{n=0}^{\infty} \frac{1}{\log_{2}(2^{2^{n}}+1)}\\ &< \frac{A}{\ln 2} \sum_{n=0}^{\infty} 2^{-n} \\ &= \frac{2A}{\ln 2}.\end{align}</math>
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| It should be stressed that this argument is in no way a rigorous [[mathematical proof|proof]]. For one thing, the argument assumes that Fermat numbers behave "[[randomness|randomly]]", yet we have already seen that the factors of Fermat numbers have special properties. If (more sophisticatedly) we regard the ''conditional'' probability that ''n'' is prime, given that we know all its prime factors exceed ''B'', as at most ''A''ln(''B'')/ln(''n''), then using Euler's theorem that the least prime factor of ''F''<sub>''n''</sub> exceeds {{nowrap|2<sup>''n'' + 1</sup>}}, we would find instead
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| : <math>\begin{align}A \sum_{n=0}^{\infty} \frac{\ln 2^{n+1}}{\ln F_{n}} &= A \sum_{n=0}^{\infty} \frac{\log_2 2^{n+1}}{\log_{2}(2^{2^{n}}+1)} \\ &<
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| A \sum_{n=0}^{\infty} (n+1) 2^{-n} \\ &= 4A.\end{align}</math>
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| Although such arguments engender the belief that there are only finitely many Fermat primes, one can also produce arguments for the opposite conclusion. Suppose we regard the conditional probability that ''n'' is prime, given that we know all its prime factors are 1 modulo ''M'', as at least ''CM''/ln(''n''). Then using Euler's result that ''M'' = 2<sup>''n'' + 1</sup> we would find that the expected total number of Fermat primes was at least
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| : <math>\begin{align}C \sum_{n=0}^{\infty} \frac{2^{n+1}}{\ln F_{n}} &= \frac{C}{\ln 2} \sum_{n=0}^{\infty} \frac{2^{n+1}}{\log_{2}(2^{2^{n}}+1)} \\ &> \frac{C}{\ln 2} \sum_{n=0}^{\infty} 1 \\ &= \infty,\end{align}</math>
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| and indeed this argument predicts that an asymptotically ''constant fraction'' of Fermat numbers are prime. | |
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| ===Equivalent conditions of primality===
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| There are a number of conditions that are [[if and only if|equivalent]] to the primality of ''F''<sub>''n''</sub>.
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| *'''[[Proth's theorem]]''' (1878)—Let ''N'' = ''k''2<sup>''m''</sup> + 1 with odd ''k'' < 2<sup>''m''</sup>. If there is an integer ''a'' such that
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| :: <math>a^{(N-1)/2} \equiv -1\pmod{N}\!</math>
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| :then ''N'' is prime. Conversely, if the above congruence does not hold, and in addition
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| :: <math>\left(\frac{a}{N}\right)=-1\!</math> (See [[Jacobi symbol]])
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| :then ''N'' is composite. If ''N'' = ''F''<sub>''n''</sub> > 3, then the above Jacobi symbol is always equal to −1 for ''a'' = 3, and this special case of Proth's theorem is known as [[Pépin's test]]. Although Pépin's test and Proth's theorem have been implemented on computers to prove the compositeness of some Fermat numbers, neither test gives a specific nontrivial factor. In fact, no specific prime factors are known for ''n'' = 20 and 24.
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| *Let ''n'' ≥ 3 be a positive odd integer. Then ''n'' is a Fermat prime if and only if for every ''a'' co-prime to ''n'', ''a'' is a [[primitive root modulo n|primitive root]] modulo ''n'' if and only if ''a'' is a [[quadratic residue|quadratic nonresidue]] modulo ''n''.
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| *The Fermat number ''F''<sub>''n''</sub> > 3 is prime if and only if it can be written uniquely as a sum of two nonzero squares, namely
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| :: <math>F_{n}=\left(2^{2^{n-1}}\right)^{2}+1^{2}.\!</math>
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| :When <math>F_{n} = x^2 + y^2</math> not of the form shown above, a proper factor is:
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| :: <math>\gcd(x + 2^{2^{n-1}} y, F_{n}).\!</math>
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| :Example 1: ''F''<sub>5</sub> = 62264<sup>2</sup> + 20449<sup>2</sup>, so a proper factor is
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| ::<math>\gcd(62264\, +\, 2^{2^4}\times 20449,\, F_{5}) = 641.\!</math>
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| :Example 2: ''F''<sub>6</sub> = 4046803256<sup>2</sup> + 1438793759<sup>2</sup>, so a proper factor is
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| ::<math>\gcd(4046803256\, +\, 2^{2^5}\times 1438793759,\, F_{6}) = 274177.\!</math>
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| ==Factorization of Fermat numbers==
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| Because of the size of Fermat numbers, it is difficult to factorize or to prove primality of those. [[Pépin's test]] gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The [[elliptic curve method]] is a fast method for finding small prime divisors of numbers. Distributed computing project ''Fermatsearch'' has successfully found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. [[Édouard Lucas]], improving the above mentioned result by Euler, proved in 1878 that every factor of Fermat number <math>F_n</math>, with ''n'' at least 2, is of the form <math>k\times2^{n+2}+1</math> (see [[Proth number]]), where ''k'' is a positive integer; this is in itself almost sufficient to prove the primality of the known Fermat primes.
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| Factorizations of the first twelve Fermat numbers are:
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| {|
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| |- valign="top"
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| |''F''<sub>0</sub> ||=|| 2<sup>1</sup>||+||1 ||=||3 is prime||
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| |- valign="top"
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| |''F''<sub>1</sub> ||=|| 2<sup>2</sup>||+||1 ||=||5 is prime||
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| |- valign="top"
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| |''F''<sub>2</sub> ||=|| 2<sup>4</sup>||+||1 ||=||17 is prime||
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| |- valign="top"
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| |''F''<sub>3</sub> ||=|| 2<sup>8</sup>||+||1 ||=||[[257 (number)|257]] is prime||
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| |- valign="top"
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| |''F''<sub>4</sub> ||=|| 2<sup>16</sup>||+||1 ||=||[[65537|65,537]] is the largest known Fermat prime||
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| |- valign="top"
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| |''F''<sub>5</sub> ||=|| 2<sup>32</sup>||+||1 ||=||4,294,967,297||
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| |- style="background:white; color:red"
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| | || || || || ||=||641 × 6,700,417
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| |- valign="top"
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| |''F''<sub>6</sub> ||=|| 2<sup>64</sup>||+||1 ||=||18,446,744,073,709,551,617||
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| |- style="background:white; color:red"
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| | || || || || ||=||274,177 × 67,280,421,310,721
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| |- valign="top"
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| |''F''<sub>7</sub> ||=|| 2<sup>128</sup>||+||1 ||=||340,282,366,920,938,463,463,374,607,431,768,211,457||
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| |- style="background:white; color:red"
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| | || || || || ||=||59,649,589,127,497,217 × 5,704,689,200,685,129,054,721
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| |-
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| |''F''<sub>8</sub> ||=|| 2<sup>256</sup>||+||1 ||=||115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,937||
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| |- style="background:white; color:red"
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| | || || || || ||=||1,238,926,361,552,897 × 93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321
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| |- valign="top"
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| |''F''<sub>9</sub> ||=|| 2<sup>512</sup>||+||1 ||=||13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,<br>976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,097||
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| |- style="background:white; color:red; vertical-align:top"
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| | || || || || ||=|| 2,424,833 × 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 × 741,640,062,627,530,801,524,787,141,901,937,474,059,940,781,097,519,023,905,821,316,144,415,759,504,705,008,092,818,711,693,940,737
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| |- valign="top"
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| |''F''<sub>10</sub> ||=|| 2<sup>1024</sup>||+||1
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| ||=||179,769,313,486,231,590,772,930,519,078,902,473,361,797,697,894,230,657,273,430,081,157,732,675,805,500,963,132,708,477,322,407,536,021,120,<br>113,879,871,393,357,658,789,768,814,416,622,492,847,430,639,474,124,377,767,893,424,865,485,276,302,219,601,246,094,119,453,082,952,085,<br>005,768,838,150,682,342,462,881,473,913,110,540,827,237,163,350,510,684,586,298,239,947,245,938,479,716,304,835,356,329,624,224,137,217||
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| |- style="background:white; color:red; vertical-align:top"
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| | || || || || ||=||45,592,577 × 6,487,031,809 × 4,659,775,785,220,018,543,264,560,743,076,778,192,897 × 130,439,874,405,488,189,727,484,768,796,509,903,946,608,530,841,611,892,186,895,295,776,832,416,251,471,863,574,<br>140,227,977,573,104,895,898,783,928,842,923,844,831,149,032,913,798,729,088,601,617,946,094,119,449,010,595,906,<br>710,130,531,906,171,018,354,491,609,619,193,912,488,538,116,080,712,299,672,322,806,217,820,753,127,014,424,577
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| |- valign="top"
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| |''F''<sub>11</sub> ||=|| 2<sup>2048</sup>||+||1
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| ||=||32,317,006,071,311,007,300,714,876,688,669,951,960,444,102,669,715,484,032,130,345,427,524,655,138,867,890,893,197,201,411,522,913,463,688,717,<br>960,921,898,019,494,119,559,150,490,921,095,088,152,386,448,283,120,630,877,367,300,996,091,750,197,750,389,652,106,796,057,638,384,067,<br>568,276,792,218,642,619,756,161,838,094,338,476,170,470,581,645,852,036,305,042,887,575,891,541,065,808,607,552,399,123,930,385,521,914,<br>333,389,668,342,420,684,974,786,564,569,494,856,176,035,326,322,058,077,805,659,331,026,192,708,460,314,150,258,592,864,177,116,725,943,<br>603,718,461,857,357,598,351,152,301,645,904,403,697,613,233,287,231,227,125,684,710,820,209,725,157,101,726,931,323,469,678,542,580,656,<br>697,935,045,997,268,352,998,638,215,525,166,389,437,335,543,602,135,433,229,604,645,318,478,604,952,148,193,555,853,611,059,596,230,657||
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| |- style="background:white; color:red; vertical-align:top"
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| | || || || || ||=|| 319,489 × 974,849 × 167,988,556,341,760,475,137 × 3,560,841,906,445,833,920,513 × 173,462,447,179,147,555,430,258,970,864,309,778,377,421,844,723,664,084,649,347,019,061,363,579,192,879,108,857,591,038,330,408,837,177,983,810,868,451,<br>546,421,940,712,978,306,134,189,864,280,826,014,542,758,708,589,243,873,685,563,973,118,948,869,399,158,545,506,611,147,420,216,132,557,017,260,564,139,<br>394,366,945,793,220,968,665,108,959,685,482,705,388,072,645,828,554,151,936,401,912,464,931,182,546,092,879,815,733,057,795,573,358,504,982,279,280,090,<br>942,872,567,591,518,912,118,622,751,714,319,229,788,100,979,251,036,035,496,917,279,912,663,527,358,783,236,647,193,154,777,091,427,745,377,038,294,<br>584,918,917,590,325,110,939,381,322,486,044,298,573,971,650,711,059,244,462,177,542,540,706,913,047,034,664,643,603,491,382,441,723,306,598,834,177
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| |-
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| |}
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| {{As of|2012|02}}, only ''F''<sub>0</sub> to ''F''<sub>11</sub> have been completely [[integer factorization|factored]].<ref name="Keller">{{Citation |first=Wilfrid |last=Keller |url=http://www.prothsearch.net/fermat.html#Summary |title=Prime Factors of Fermat Numbers |work=ProthSearch.net |date=February 7, 2012|accessdate=March 5, 2012}}</ref> The [[distributed computing]] project Fermat Search is searching for new factors of Fermat numbers.<ref>[http://www.fermatsearch.org/ FermatSearch.org]</ref> The set of all Fermat factors is [[OEIS:A050922|A050922]] (or, sorted, [[OEIS:A023394|A023394]]) in [[On-Line Encyclopedia of Integer Sequences|OEIS]].
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| ==Pseudoprimes and Fermat numbers==
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| Like [[composite number]]s of the form 2<sup>''p''</sup> − 1, every composite Fermat number is a [[strong pseudoprime]] to base 2. Because ''all'' strong pseudoprimes to base 2 are also Fermat [[pseudoprimes]] - i.e.
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| :<math>2^{F_n-1} \equiv 1 \pmod{F_n}\,\!</math>
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| for all Fermat numbers.
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| It is generally believed that all but the first few Fermat numbers are composite. If proven true, this would mean it is possible to generate infinitely many strong pseudoprimes to base 2 from the Fermat numbers.
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| In 1964, Rotkiewicz showed that the product of any number of prime '''''or''''' composite Fermat numbers will be a Fermat pseudoprime to the base 2.
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| ==Other theorems about Fermat numbers==
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| '''Lemma: If ''n'' is a positive integer, '''
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| :<math>a^n-b^n=(a-b)\sum_{k=0}^{n-1} a^kb^{n-1-k}.</math>
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| ''proof:''
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| :<math>(a-b)\sum_{k=0}^{n-1}a^kb^{n-1-k}</math>
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| :<math>=\sum_{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum_{k=0}^{n-1}a^kb^{n-k}</math>
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| :<math>=a^n+\sum_{k=1}^{n-1}a^kb^{n-k}-\sum_{k=1}^{n-1}a^kb^{n-k}-b^n</math>
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| :<math>=a^n-b^n.</math>
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| '''Theorem: If <math>2^n+1</math> is an odd prime, then <math>n</math> is a power of 2.'''
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| ''proof:''
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| If <math>n</math> is a positive integer but not a power of 2, then <math>n = rs</math> where
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| <math>1 \le r < n</math>, <math>1 < s \le n</math> and <math>s</math> is odd.
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| By the preceding lemma, for positive integer <math>m</math>,
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| :<math>(a-b) \mid (a^m-b^m)</math>
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| where <math> \mid </math> means "evenly divides". Substituting <math>a = 2^r</math>, <math>b = -1</math>, and <math>m = s</math> and using that <math> s </math> is odd,
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| :<math> (2^r+1) \mid (2^{rs}+1), </math>
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| and thus
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| :<math> (2^r+1) \mid (2^n+1). </math>
| |
| | |
| Because <math>1 < 2^r+1 < 2^n+1</math>, it follows that <math>2^n+1</math> is not prime. Therefore, by contraposition <math>n</math> must be a power of 2.
| |
| | |
| '''Theorem: A Fermat prime cannot be a [[Wieferich prime]].'''
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| | |
| Proof: We show if <math>p=2^m+1</math> is a Fermat prime, then the congruence <math>2^{p-1} \equiv 1 \pmod {p^2}</math>
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| does not satisfy.
| |
| | |
| It is easy to show
| |
| <math>2m |p-1</math>. Now write, <math>p-1=2m\lambda</math>. If the given congruence satisfies, then <math>p^2|2^{2m\lambda}-1</math>, therefore
| |
| | |
| :<math>0 \equiv (2^{2m\lambda}-1)/(2^m+1)=(2^m-1)(1+2^{2m}+2^{4m}+...+2^{2(\lambda-1)m}) \equiv -2\lambda \pmod {2^m+1}.\ </math>
| |
| | |
| Hence <math>2^m+1|2\lambda</math>,and therefore
| |
| <math>2\lambda \geq 2^m+1</math>. This leads to
| |
| | |
| <math>p-1 \geq m(2^m+1)</math>, which is impossible since <math>m \geq 2</math>.
| |
| | |
| '''A theorem of [[Édouard Lucas]]: Any prime divisor ''p'' of ''F''<sub>n</sub> = <math> 2^{2^{\overset{n}{}}}+1</math> is of the form <math>k2^{n+2}+1</math> whenever n is greater than one.'''
| |
| | |
| ''Sketch of proof:''
| |
| | |
| Let ''G''<sub>''p''</sub> denote the group of non-zero elements of the integers (mod ''p'') under multiplication, which has order ''p-1''. Notice that ''2'' (strictly speaking, its image (mod ''p'')) has multiplicative order <math>2^{n+1}</math> in ''G''<sub>''p''</sub>, so that, by [[Lagrange's theorem (group theory)|Lagrange's theorem]], ''p-1'' is divisible by <math>2^{n+1} </math> and ''p'' has the form <math>k2^{n+1}+1</math> for some integer ''k'',
| |
| as [[Euler]] knew. Édouard Lucas went further. Since ''n'' is greater than ''1'', the prime ''p'' above is congruent to 1 (mod ''8''). Hence (as was known to [[Carl Friedrich Gauss]]), ''2'' is a [[quadratic residue]] (mod ''p''), that is, there is integer ''a'' such that ''a''<sup>2</sup> -2 is divisible by ''p''. Then the image of ''a'' has order <math>2^{n+2}</math> in the group ''G''<sub>''p''</sub> and (using Lagrange's theorem again), ''p-1'' is divisible by <math>2^{n+2}</math>
| |
| and ''p'' has the form <math>s2^{n+2}+1</math> for some integer ''s''.
| |
| | |
| In fact, it can be seen directly that ''2'' is a quadratic residue (mod ''p''), since
| |
| <math>(1 +2^{2^{n-1}})^{2} \equiv 2^{1+2^{n-1}}</math> (mod ''p''). Since an
| |
| odd power of ''2'' is a quadratic residue (mod ''p''), so is ''2'' itself.
| |
| | |
| ==Relationship to constructible polygons==
| |
| {{main|Constructible polygon}}
| |
| | |
| An ''n''-sided regular polygon can be constructed with [[compass and straightedge]] if and only if ''n'' is the product of a power of 2 and distinct Fermat primes. In other words, if and only if ''n'' is of the form ''n'' = 2<sup>''k''</sup>''p''<sub>1</sub>''p''<sub>2</sub>…''p''<sub>''s''</sub>, where ''k'' is a nonnegative integer and the ''p''<sub>''i''</sub> are distinct Fermat primes.
| |
| | |
| A positive integer ''n'' is of the above form if and only if its [[Euler's totient function|totient]] φ(''n'') is a power of 2.
| |
| | |
| ==Applications of Fermat numbers==
| |
| | |
| ===Pseudorandom Number Generation===
| |
| Fermat primes are particularly useful in generating pseudo-random sequences of numbers in the range 1 … ''N'', where ''N'' is a power of 2. The most common method used is to take any seed value between 1 and ''P'' − 1, where ''P'' is a Fermat prime. Now multiply this by a number ''A'', which is greater than the square root of ''P'' and is a [[primitive root modulo n|primitive root]] modulo ''P'' (i.e., it is not a [[quadratic residue]]). Then take the result modulo ''P''. The result is the new value for the RNG.
| |
| : <math>V_{j+1} = \left( A \times V_j \right) \bmod P</math> (see [[Linear congruential generator]], [[RANDU]])
| |
| This is useful in computer science since most data structures have members with 2<sup>''X''</sup> possible values. For example, a byte has 256 (2<sup>8</sup>) possible values (0–255). Therefore to fill a byte or bytes with random values a random number generator which produces values 1–256 can be used, the byte taking the output value − 1. Very large Fermat primes are of particular interest in data encryption for this reason. This method produces only [[pseudorandom]] values as, after ''P'' − 1 repetitions, the sequence repeats. A poorly chosen multiplier can result in the sequence repeating sooner than ''P'' − 1.
| |
| | |
| ==Other interesting facts==
| |
| A Fermat number cannot be a perfect number or part of a pair of [[amicable numbers]]. {{harv|Luca|2000}}
| |
| | |
| The series of reciprocals of all prime divisors of Fermat numbers is [[Convergent series|convergent]]. {{harv|Křížek|Luca|Somer|2002}}
| |
| | |
| If ''n''<sup>''n''</sup> + 1 is prime, there exists an integer ''m'' such that ''n'' = 2<sup>2</sup><sup><sup>''m''</sup></sup>. The equation
| |
| ''n''<sup>''n''</sup> + 1 = ''F''<sub>(2</sub><sub><sup>''m''</sup></sub><sub>+''m'')</sub>
| |
| holds at that time.<ref>Jeppe Stig Nielsen, [http://jeppesn.dk/nton.html "S(n) = n^n + 1"].</ref>
| |
| | |
| Let the largest prime factor of Fermat number ''F''<sub>''n''</sub> be ''P''(''F''<sub>''n''</sub>). Then,
| |
| :<math>P(F_n)\ge 2^{n+2}(4n+9)+1.</math> {{harv|Grytczuk|Luca|Wójtowicz|2001}}
| |
| | |
| ==Generalized Fermat numbers==
| |
| Numbers of the form <math>a^{2^{ \overset{n} {}}} + b^{2^{ \overset{n} {}}}</math>, where ''a'' > 1 are called '''generalized Fermat numbers'''. An odd prime ''p'' is a generalized Fermat number if and only if ''p'' is congruent to [[Pythagorean prime|1 (mod 4)]]. (Here we consider only the case ''n''>0, so 3 = <math>2^{2^{0}}+1</math> is not a counterexample.)
| |
| | |
| By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form <math>a^{2^{ \overset{n} {}}} + 1</math> as ''F<sub>n</sub>''(''a''). In this notation, for instance, the number 100,000,001 would be written as ''F''<sub>3</sub>(10). In the following we shall restrict ourselves to primes of this form, <math>a^{2^{ \overset{n} {}}} + 1</math>.
| |
| | |
| If we require ''n''>0, then [[Landau's problems|Landau's fourth problem]] asks if there are infinitely many generalized Fermat primes ''F<sub>n</sub>''(''a'').
| |
| | |
| ===Generalized Fermat primes===
| |
| Because of the ease of proving their primality, generalized Fermat primes have become in recent years a hot topic for research within the field of number theory. Many of the largest known primes today are generalized Fermat primes.
| |
| | |
| Generalized Fermat numbers can be prime only for [[Even and odd numbers|even]] ''a'', because if ''a'' is odd then every generalized Fermat number will be divisible by 2. By analogy with the [[heuristic argument]] for the finite number of primes among the base-2 Fermat numbers, it is to be expected that there will be only finitely many generalized Fermat primes for each even base. The smallest prime number ''F<sub>n</sub>''(''a'') with ''n'' > 4 is ''F''<sub>5</sub>(30), or 30<sup>32</sup>+1.
| |
| | |
| A more elaborate theory can be used to predict the number of bases for which ''F<sub>n</sub>''(''a'') will be prime for a fixed ''n''. The number of generalized Fermat primes can be roughly expected to halve as ''n'' is increased by 1.
| |
| | |
| ===Largest known generalized Fermat primes===
| |
| The following is a list of the 10 largest known generalized Fermat primes.<ref name="Top Twenty's Generalized Fermat Primes">{{cite web|title=Top Twenty's Generalized Fermat Primes|url=http://primes.utm.edu/top20/page.php?id=12|work=The Prime Pages|first=Chris K.|last=Caldwell|accessdate=15 July 2013}}</ref> They are all [[megaprime]]s. {{As of|2013|7}} the whole top-10 was discovered by participants in the [[PrimeGrid]] project.
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! Rank
| |
| ! Largest known prime rank<ref>{{cite web|title=Search on Generalized Fermat|url=http://primes.utm.edu/primes/search.php?Number=10&Comment=Generalized%20Fermat|work=The Prime Pages|first=Chris K.|last=Caldwell|accessdate=15 July 2013}}</ref>
| |
| ! Prime number
| |
| ! Generalized Fermat notation
| |
| ! Number of digits
| |
| ! Found date
| |
| ! reference
| |
| |-
| |
| | 1
| |
| | 12
| |
| | 475856<sup>524288</sup> + 1
| |
| | ''F<sub>19</sub>''(475856)
| |
| | 2,976,633
| |
| | 2012 August 8
| |
| | <ref name="PrimeGrid's Generalized Fermat Prime Search - 475856^524288+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 475856^524288+1|url=http://www.primegrid.com/download/GFN-475856_524288.pdf|publisher=Primegrid|accessdate=21 August 2012}}</ref>
| |
| |-
| |
| | 2
| |
| | 13
| |
| | 356926<sup>524288</sup> + 1
| |
| | ''F<sub>19</sub>''(356926)
| |
| | 2,911,151
| |
| | 2012 June 20
| |
| |<ref name="PrimeGrid's Generalized Fermat Prime Search - 356926^524288+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 356926^524288+1|url=http://www.primegrid.com/download/GFN-356926_524288.pdf|publisher=Primegrid|accessdate=30 July 2012}}</ref>
| |
| |-
| |
| | 3
| |
| | 14
| |
| | 341112<sup>524288</sup> + 1
| |
| | ''F<sub>19</sub>''(341112)
| |
| | 2,900,832
| |
| | 2012 June 15
| |
| |<ref name="PrimeGrid's Generalized Fermat Prime Search - 341112^524288+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 341112^524288+1|url=http://www.primegrid.com/download/GFN-341112_524288.pdf|publisher=Primegrid|accessdate=9 July 2012}}</ref>
| |
| |-
| |
| | 4
| |
| | 17
| |
| | 75898<sup>524288</sup> + 1
| |
| | ''F<sub>19</sub>''(75898)
| |
| | 2,558,647
| |
| | 2011 November 19
| |
| | <ref name="PrimeGrid's Generalized Fermat Prime Search - 75898^524288+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 75898^524288+1|url=http://www.primegrid.com/download/gfn-75898_524288.pdf|publisher=Primegrid|accessdate=9 July 2012}}</ref>
| |
| |-
| |
| | 5
| |
| | 28
| |
| | 773620<sup>262144</sup> + 1
| |
| | ''F<sub>18</sub>''(773620)
| |
| | 1,543,643
| |
| | 2012 April 19
| |
| | <ref name="PrimeGrid's Generalized Fermat Prime Search - 773620^262144+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 773620^262144+1|url=http://www.primegrid.com/download/gfn-773620_262144.pdf|publisher=Primegrid|accessdate=9 July 2012}}</ref>
| |
| |-
| |
| | 6
| |
| | 30
| |
| | 676754<sup>262144</sup> + 1
| |
| | ''F<sub>18</sub>''(676754)
| |
| | 1,528,413
| |
| | 2012 February 12
| |
| | <ref name="PrimeGrid's Generalized Fermat Prime Search - 676754^262144+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 676754^262144+1|url=http://www.primegrid.com/download/gfn-676754_262144.pdf|publisher=Primegrid|accessdate=9 July 2012}}</ref>
| |
| |-
| |
| | 7
| |
| | 32
| |
| | 525094<sup>262144</sup> + 1
| |
| | ''F<sub>18</Sub>''(525094)
| |
| | 1,499,526
| |
| | 2012 January 18
| |
| | <ref name="PrimeGrid's Generalized Fermat Prime Search - 525094^262144+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 525094^262144+1|url=http://www.primegrid.com/download/gfn-525094_262144.pdf|publisher=Primegrid|accessdate=9 July 2012}}</ref>
| |
| |-
| |
| | 8
| |
| | 35
| |
| | 361658<sup>262144</sup> + 1
| |
| | ''F<sub>18</sub>''(361658)
| |
| | 1,457,075
| |
| | 2011 October 29
| |
| | <ref name="PrimeGrid's Generalized Fermat Prime Search - 361658^262144+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 361658^262144+1|url=http://www.primegrid.com/download/gfn-361658_262144.pdf|publisher=Primegrid|accessdate=9 July 2012}}</ref>
| |
| |-
| |
| | 9
| |
| | 38
| |
| | 145310<sup>262144</sup> + 1
| |
| | ''F<sub>18</sub>''(145310)
| |
| | 1,353,265
| |
| | 2011 February 8
| |
| | <ref name="PrimeGrid's Generalized Fermat Prime Search - 145310^262144+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 145310^262144+1|url=http://www.primegrid.com/download/gfn-145310_262144.pdf|publisher=Primegrid|accessdate=9 July 2012}}</ref>
| |
| |-
| |
| | 10
| |
| | 44
| |
| | 40734<sup>262144</sup> + 1
| |
| | ''F<sub>18</Sub>''(40734)
| |
| | 1,208,473
| |
| | 2011 March 8
| |
| | <ref name="PrimeGrid's Generalized Fermat Prime Search - 40734^262144+1">{{cite web|title=PrimeGrid’s Generalized Fermat Prime Search - 40734^262144+1|url=http://www.primegrid.com/download/gfn-40734_262144.pdf|publisher=Primegrid|accessdate=9 July 2012}}</ref>
| |
| |-
| |
| |}
| |
| | |
| ==See also==
| |
| * [[Constructible polygon]]: which regular polygons are constructible partially depends on Fermat primes.
| |
| * [[Double exponential function]]
| |
| * [[Lucas' theorem]]
| |
| * [[Mersenne prime]]
| |
| * [[Pierpont prime]]
| |
| * [[Primality test]]
| |
| * [[Proth's theorem]]
| |
| * [[Pseudoprime]]
| |
| * [[Sierpiński number]]
| |
| * [[Sylvester's sequence]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{Citation |last=Golomb |first=S. W. |year=1963 |date=January 1, 1963 |title=On the sum of the reciprocals of the Fermat numbers and related irrationalities |journal=Canadian Journal of Mathematics |volume=15 |issue= |publisher=Canadian Mathematical Society |pages=475–478 |doi=10.4153/CJM-1963-051-0 |url=http://cms.math.ca/10.4153/CJM-1963-051-0 |format=PDF |issn=0008-414X }}
| |
| *{{Citation |last=Grytczuk |first=A. |last2=Luca |first2=F. |last3=Wójtowicz |first3=M. |lastauthoramp=yes |year=2001 |title=Another note on the greatest prime factors of Fermat numbers |journal=Southeast Asian Bulletin of Mathematics |volume=25 |issue=1 |publisher=Springer-Verlag |pages=111–115 |doi=10.1007/s10012-001-0111-4 |url=http://link.springer.com/article/10.1007/s10012-001-0111-4 |issn=0129-2021 }}
| |
| *{{citation |last=Guy |first=Richard K. |authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory |year=2004 |edition=3rd |publisher=[[Springer Verlag]] |series=Problem Books in Mathematics |volume=1 |location=New York |isbn=0-387-20860-7 |pages=A3, A12, B21 |url=http://www.springer.com/mathematics/numbers/book/978-0-387-20860-2?otherVersion=978-0-387-26677-0 }}
| |
| *{{citation |last=Křížek |first=Michal |last2=Luca |first2=Florian |last3=Somer |first3=Lawrence |lastauthoramp=yes |chapter= |title=17 Lectures on Fermat Numbers: From Number Theory to Geometry |year=2001 |series=CMS books in mathematics |volume=10 |publisher=Springer |location=New York |isbn=0-387-95332-9 |pages= |url=http://www.springer.com/mathematics/numbers/book/978-0-387-95332-8 }} - This book contains an extensive list of references.
| |
| *{{Citation |last=Křížek |first=Michal |last2=Luca |first2=Florian |last3=Somer |first3=Lawrence |lastauthoramp=yes |year=2002 |title=On the convergence of series of reciprocals of primes related to the Fermat numbers |journal=Journal of Number Theory |volume=97 |issue=1 |publisher=Elsevier |pages=95–112 |doi=10.1006/jnth.2002.2782 |url=http://www.sciencedirect.com/science/journal/0022314X/97/1 |format=PDF |issn=0022-314X }}
| |
| *{{Citation |last=Luca |first=Florian |year=2000 |title=The anti-social Fermat number |journal=American Mathematical Monthly |volume=107 |issue=2 |publisher=Mathematical Association of America |pages=171–173 |doi=10.2307/2589441 |url=http://www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-february-2000 |issn=0002-9890 }}
| |
| *{{Citation |last=Ribenboim |first=Paulo |author-link=Paulo Ribenboim |year=1996 |title=The New Book of Prime Number Records |publisher=Springer |location=New York |edition=3rd |isbn=0-387-94457-5 |url=http://www.springer.com/mathematics/numbers/book/978-0-387-94457-9 }}
| |
| *{{Citation |last=Robinson |first=Raphael M. |title=Mersenne and Fermat Numbers |journal=Proceedings of the American Mathematical Society |volume=5 |issue=5 |year=1954 |publisher=American Mathematical Society |pages=842–846 |doi=10.2307/2031878 }}
| |
| *{{citation |last=Yabuta |first=M. |journal=Fibonacci Quarterly |publisher= Fibonacci Association |pages=439–443 |title=A simple proof of Carmichael's theorem on primitive divisors |url=http://www.fq.math.ca/Scanned/39-5/yabuta.pdf |format=PDF |volume=39 |year=2001 |issn=0015-0517 }}
| |
| | |
| ==External links==
| |
| * Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=FermatNumber The Prime Glossary: Fermat number] at The [[Prime Pages]].
| |
| * Luigi Morelli, [http://www.fermatsearch.org/history.html History of Fermat Numbers]
| |
| * John Cosgrave, [http://www.spd.dcu.ie/johnbcos/fermat6.htm Unification of Mersenne and Fermat Numbers]
| |
| * Wilfrid Keller, [http://www.prothsearch.net/fermat.html Prime Factors of Fermat Numbers]
| |
| * {{MathWorld|title=Fermat Number|urlname=FermatNumber}}
| |
| * {{MathWorld|title=Fermat Prime|urlname=FermatPrime}}
| |
| * {{MathWorld|title=Fermat Pseudoprime|urlname=FermatPseudoprime}}
| |
| * {{MathWorld|title=Generalized Fermat Number|urlname=GeneralizedFermatNumber}}
| |
| * Yves Gallot, [http://pagesperso-orange.fr/yves.gallot/primes/index.html Generalized Fermat Prime Search]
| |
| * Mark S. Manasse, [http://www.google.com/groups?selm=1990Jun15.190100.8505%40src.dec.com&oe=UTF-8&output=gplain Complete factorization of the ninth Fermat number] (original announcement)
| |
| * Peyton Hayslette, [http://www.primegrid.com/download/GFN-341112_524288.pdf Largest Known Generalized Fermat Prime Announcement]
| |
| | |
| {{Prime number classes|state=collapsed}}
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| {{Classes of natural numbers}}
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| {{DEFAULTSORT:Fermat Number}}
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| [[Category:Number theory]]
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| [[Category:Euclidean plane geometry]]
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| [[Category:Articles containing proofs]]
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| [[Category:Unsolved problems in mathematics]]
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| [[Category:Large integers]]
| |
| [[Category:Classes of prime numbers]]
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| [[Category:Integer sequences]]
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| [[da:Fermatprimtal]]
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