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* ''17 Lectures on Fermat Numbers: From Number Theory to Geometry'', Michal Krizek, Florian Luca, Lawrence Somer, Springer, CMS Books 9, ISBN 0387953329 (This book contains an extensive list of references.)  * ''17 Lectures on Fermat Numbers: From Number Theory to Geometry'', Michal Krizek, Florian Luca, Lawrence Somer, Springer, CMS Books 9, ISBN 0387953329 (This book contains an extensive list of references.)  
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Revision as of 11:09, 1 July 2020
In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form
 [math]F_{n} = 2^{2^n} + 1[/math]
where n is a nonnegative integer. The first eight Fermat numbers are A000215:
 F_{0} = 2^{1} + 1 = 3
 F_{1} = 2^{2} + 1 = 5
 F_{2} = 2^{4} + 1 = 17
 F_{3} = 2^{8} + 1 = 257
 F_{4} = 2^{16} + 1 = 65537
 F_{5} = 2^{32} + 1 = 4294967297 = 641 × 6700417
 F_{6} = 2^{64} + 1 = 18446744073709551617 = 274177 × 67280421310721
 F_{7} = 2^{128} + 1 = 340282366920938463463374607431768211457 = 59649589127497217 × 5704689200685129054721
Only the first 12 Fermat numbers have been completely factored. These factorisations can be found at Prime Factors of Fermat Numbers
If n is a positive integer and 2^{n} + 1 is prime, it can be shown that n must be a power of 2. (If n = ab where 1 < a, b < n and b is odd, then 2^{n} + 1 ≡ (2^{a})^{b} + 1 ≡ (−1)^{b} + 1 ≡ 0 (mod 2^{a} + 1).) In other words, every prime of the form 2^{n} + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F_{0},...,F_{4}.
Contents
Basic properties
The Fermat numbers satisfy the following recurrence relations
 [math]F_{n} = (F_{n1}1)^{2}+1[/math]
 [math]F_{n} = F_{n1} + 2^{2^{n1}}F_{0} \cdots F_{n2}[/math]
 [math]F_{n} = F_{n1}^2  2(F_{n2}1)^2[/math]
 [math]F_{n} = F_{0} \cdots F_{n1} + 2[/math]
for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor. To see this, suppose that 0 ≤ i < j and F_{i} and F_{j} have a common factor a > 1. Then a divides both
 [math]F_{0} \cdots F_{j1}[/math]
and F_{j}; 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 infinitude of the prime numbers: for each F_{n}, choose a prime factor p_{n}; then the sequence {p_{n}} is an infinite sequence of distinct primes.
Here are some other basic properties of the Fermat numbers:
 If n ≥ 2, then F_{n} ≡ 17 or 41 (mod 72). (See modular arithmetic)
 If n ≥ 2, then F_{n} ≡ 17, 37, 57, or 97 (mod 100).
 The number of digits D(n,b) of F_{n} expressed in the base b is
 [math]D(n,b) = \lfloor \log_{b}\left(2^{2^{n}}+1\right)+1 \rfloor \approx \lfloor 2^{n}\,\log_{b}2+1 \rfloor [/math] (See floor function)
 No Fermat number can be expressed as the sum of two primes, with the exception of F_{1} = 2 + 3.
 No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime.
Primality of Fermat numbers
Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the first five Fermat numbers F_{0},...,F_{4} are easily shown to be prime. However, this conjecture was refuted by Leonhard Euler in 1732 when he showed that
 [math]F_{5} = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297 = 641 \cdot 6700417[/math]
 See Euler's work and Fermat's two letters. Original document and English translation is available.
It is interesting to note how Euler found this factorization. Euler had proved that every factor of F_{n} must have the form k2^{n+1} + 1. For n = 5, this means that the only possible factors are of the form 64k + 1. It did not take Euler very long to find the factor 641 = 10×64 + 1. It isn't hard to check this, but there is an interesting proof for this: [math]641=2^4+5^4=5*2^7+1[/math] using this we can get
 [math]GCD(2^{2^5}+1,641)=GCD(2^{4}*2^{28}+1,641)=GCD(5^{4}*2^{28}+1,641)=GCD((5*2^{7})^{4}1,641)=GCD((1)^{4}1,641)=GCD(0,641)=641[/math]
So [math]F_{5}[/math] is divisible by 641.
It is widely believed that Fermat was aware of Euler's result, so it seems curious why he failed to follow through on the straightforward calculation to find the factor. One common explanation is that Fermat made a computational mistake and was so convinced of the correctness of his claim that he failed to doublecheck his work.
There are no other known Fermat primes F_{n} with n > 4. In fact, each of the following is an open problem:
 Is F_{n} composite for all n > 4?
 Are there infinitely many Fermat primes?
 Are there infinitely many composite Fermat numbers?
 Is every Fermat numbers squarefree?
The following heuristic argument suggests there are only finitely many Fermat primes: according to the prime number theorem, the "probability" that a random number n is prime is at most A/ln(n), where A is a fixed constant. Therefore, the total expected number of Fermat primes is at most
 [math]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)} \lt \frac{A}{\ln 2} \sum_{n=0}^{\infty} 2^{n} = \frac{2A}{\ln 2}[/math]
It should be stressed that this argument is in no way a rigorous proof. For one thing, the argument assumes that Fermat numbers behave "randomly", yet we have already seen that the factors of Fermat numbers have special properties. Although it is widely believed that there are only finitely many Fermat primes, it should be noted that there are some experts who disagree. [1]
As of this writing (2004), it is known that F_{n} is composite for 5 ≤ n ≤ 32, although complete factorisations of F_{n} are known only for 0 ≤ n ≤ 11. The largest known composite Fermat number is F_{2478782}, and its prime factor 3×2^{2478785} + 1 was discovered by John Cosgrave and his ProthGallot Group on October 10, 2003.
There are a number of conditions that are equivalent to the primality of F_{n}.
 Proth's theorem  (1878) Let N = k2^{m} + 1 with odd k < 2^{m}. If there is an integer a such that
 [math]a^{(N1)/2} \equiv 1 \,\pmod N [/math]
 then N is prime. Conversely, if the above congruence does not hold, and in addition
 [math]\left(\frac{a}{N}\right)=1[/math] (See Wikipedia:Jacobi symbol)
 then N is composite. If N = F_{n} > 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 many Fermat numbers, neither test gives a specific nontrivial factor. In fact, no specific prime factors are known for n = 14, 20, 22, and 24.
 Theorem: Let n>0 then [math]F_n[/math] is prime if and only if it divides [math]S_{2^n2}[/math] where [math]S_0=5[/math] and [math]S_{k+1}=S^2_{k}2[/math] [ http://www.robertgerbicz.tar.hu/Fermatnumbers.pdf See the proof by Robert Gerbicz]
 Let n ≥ 3 be a positive odd integer. Then n is a Fermat prime if and only if for every a coprime to n, a is a primitive root mod n if and only if a is a quadratic nonresidue mod n.
 The Fermat number F_{n} > 3 is prime if and only if it can be written uniquely as a sum of two nonzero squares, namely
 [math]F_{n}=\left(2^{2^{n1}}\right)^{2}+1^{2}[/math]
 When [math]F_{n} = x^2 + y^2[/math] not of the form shown above, a proper factor is:
 [math]\gcd(x + 2^{2^{n1}} y, F_{n})[/math]
 Example 1: F_{5} = 62264^{2} + 20449^{2}, so a proper factor is [math]\gcd(62264\, +\, 2^{2^4}\, 20449,\, F_{5}) = 641[/math].
 Example 2: F_{6} = 4046803256^{2} + 1438793759^{2}, so a proper factor is [math]\gcd(4046803256\, +\, 2^{2^5}\, 1438793759,\, F_{6}) = 274177[/math].
Factorization of Fermat numbers
Because or the size of Fermat numbers, it is difficult to factorize or to prove primality of those. Pépin's test is necessary and sufficient test for primality of Fermat numbers which can be implemented by modern computers. Elliptic curve method is a fast method for finding small prime divisors of numbers, and at least GIMPS is trying to find prime divisors of Fermat numbers by elliptic curve method. Distributed computing project Fermatsearch has also succesfully found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Lucas proved in year 1878 that every factor of Fermat number [math]F_n[/math] is of the form [math]k2^{n+2}+1[/math], where k is a nonnegative integer.
 Factorization of the eighth Fermat number
 Original announcement of the factorization of the ninth Fermat number
 In 1997 Richard P. Brent rediscovered the factorization of F9 by ECM method
 Factorization of the tenth Fermat number
 Factorization of the eleventh Fermat number
Fermat's little theorem and pseudoprimes
...Using Fermat numbers to generate infinitely many pseudoprimes...
Relationship to constructible polygons
An nsided regular polygon can be constructed with ruler and compass if and only if n is a power of 2 or 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^{k}p_{1}p_{2}...p_{s}, where k is a nonnegative integer and the p_{i} are distinct Fermat primes. See constructible polygon.
A positive integer n is of the above form if and only if φ(n) is a power of 2, where φ(n) is Euler's totient function.
Applications of Fermat numbers
...Fermat number transform...random number generation...
Other interesting facts
...F_{n} cannot be a perfect power, perfect, or part of amicable pair, etc...
Generalized Fermat numbers
Main title: Generalized Fermat number ...brief definition of L(p,m) and G(p,m)...
See also
 Mersenne prime
 Lucas's theorem
 Proth's theorem
 Pseudoprime
 Primality test
 Constructible number
 Sierpiński problem
External links
 Fermat number
 Sequence of Fermat numbers
 Prime Glossary Page on Fermat Numbers
 Generalized Fermat Prime search
 History of Fermat Numbers
 Unification of Mersenne and Fermat Numbers
 Prime Factors of Fermat Numbers
References:
 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Michal Krizek, Florian Luca, Lawrence Somer, Springer, CMS Books 9, ISBN 0387953329 (This book contains an extensive list of references.)
General numbers 
Special numbers 
Prime numbers 
