Fermat number

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In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form

F_{n} = 2^{2^n} + 1

where n is a nonnegative integer. The first eight Fermat numbers are (sequence A000215 in OEIS):

F0 = 21 + 1 = 3
F1 = 22 + 1 = 5
F2 = 24 + 1 = 17
F3 = 28 + 1 = 257
F4 = 216 + 1 = 65537
F5 = 232 + 1 = 4294967297 = 641 × 6700417
F6 = 264 + 1 = 18446744073709551617 = 274177 × 67280421310721
F7 = 2128 + 1 = 340282366920938463463374607431768211457 = 59649589127497217 × 5704689200685129054721

As of 2007, only the first 12 Fermat numbers have been completely factorized. These factorizations can be found at Prime Factors of Fermat Numbers

If 2n + 1 is 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 2n + 1 ≡ (2a)b + 1 ≡ (−1)b + 1 ≡ 0 (mod 2a + 1).) In other words, every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F0,...,F4.

Contents

[edit] Basic properties

The Fermat numbers satisfy the following recurrence relations

F_{n} = (F_{n-1}-1)^{2}+1\,
F_{n} = F_{n-1} + 2^{2^{n-1}}F_{0} \cdots F_{n-2}
F_{n} = F_{n-1}^2 - 2(F_{n-2}-1)^2
F_{n} = F_{0} \cdots F_{n-1} + 2

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 Fi and Fj have a common factor a > 1. Then a divides both

F_{0} \cdots F_{j-1}

and Fj; 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 Fn, choose a prime factor pn; then the sequence {pn} is an infinite sequence of distinct primes.

Here are some other basic properties of the Fermat numbers:

  • If n ≥ 2, then Fn ≡ 17 or 41 (mod 72). (See modular arithmetic)
  • If n ≥ 2, then all Fn end in the same decimal digit 7.
  • If n ≥ 2, then Fn ≡ 17, 37, 57, or 97 (mod 100).
  • The number of digits D(n,b) of Fn expressed in the base b is
D(n,b) = \lfloor \log_{b}\left(2^{2^{n}}+1\right)+1 \rfloor \approx \lfloor 2^{n}\,\log_{b}2+1 \rfloor (See floor function)
  • No Fermat number can be expressed as the sum of two primes, with the exception of F1 = 2 + 3.
  • No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime.

[edit] 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 F0,...,F4 are easily shown to be prime. However, this conjecture was refuted by Leonhard Euler in 1732 when he showed that

F_{5} = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297 = 641 \cdot 6700417 \;

Euler proved that every factor of Fn must have the form k2n+1 + 1. For n = 5, this means that the only possible factors are of the form 64k + 1. Euler found the factor 641 = 10×64 + 1.

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 double-check his work.

There are no other known Fermat primes Fn with n > 4. In fact, each of the following is an open problem:

  • Is Fn composite for all n > 4?
  • Are there infinitely many Fermat primes?
  • Are there infinitely many composite Fermat numbers?

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. Therefore, the total expected number of Fermat primes is at most

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}

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, there are some experts who disagree. [1]

As of 2006 it is known that Fn is composite for 5 ≤ n ≤ 32, although complete factorisations of Fn are known only for 0 ≤ n ≤ 11, and there are no known factors for n in {14, 20, 22, 24}. The largest known composite Fermat number is F2478782, and its prime factor 3×22478785 + 1 was discovered by John Cosgrave and his Proth-Gallot Group on October 10, 2003. An even more speculative application of the heuristic argument above suggests - subject to the same caveats - that the "probability" that there are any new Fermat primes beyond F32 is on the order of one in a billion.

There are a number of conditions that are equivalent to the primality of Fn.

  • Proth's theorem -- (1878) Let N = k2m + 1 with odd k < 2m. If there is an integer a such that
a^{(N-1)/2} \equiv -1 \mod N
then N is prime. Conversely, if the above congruence does not hold, and in addition
\left(\frac{a}{N}\right)=-1 (See Jacobi symbol)
then N is composite. If N = Fn > 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.
  • 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 mod n if and only if a is a quadratic nonresidue mod n.
  • The Fermat number Fn > 3 is prime if and only if it can be written uniquely as a sum of two nonzero squares, namely
F_{n}=\left(2^{2^{n-1}}\right)^{2}+1^{2}
When Fn = x2 + y2 not of the form shown above, a proper factor is:
\gcd(x + 2^{2^{n-1}} y, F_{n})
Example 1: F5 = 622642 + 204492, so a proper factor is \gcd(62264\, +\, 2^{2^4}\, 20449,\, F_{5}) = 641.
Example 2: F6 = 40468032562 + 14387937592, so a proper factor is \gcd(4046803256\, +\, 2^{2^5}\, 1438793759,\, F_{6}) = 274177.

[edit] Factorization of Fermat numbers

Because of 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. The 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 successfully 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 Fn is of the form 2n + 2k + 1, where k is a positive integer.

[edit] Fermat's little theorem and pseudoprimes

Fermat's little theorem

...Using Fermat numbers to generate infinitely many pseudoprimes...

[edit] Other theorems about Fermat numbers

Lemma: If n is a positive integer,

a^n-b^n=(a-b)\sum_{k=0}^{n-1} a^kb^{n-1-k}.

proof:

(a-b)\sum_{k=0}^{n-1}a^kb^{n-1-k}
=\sum_{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum_{k=0}^{n-1}a^kb^{n-k}
=a^n+\sum_{k=1}^{n-1}a^kb^{n-k}-\sum_{k=1}^{n-1}a^kb^{n-k}-b^n
= an − bn

Theorem: If 2n + 1 is prime, then n is zero or a power of 2.

proof:

For n = 0, 20 + 1 equals prime number 2.

If n is a positive integer but not a power of 2, then n = rs where 1 \le r < n, 1 < s \le n and s is odd.

By the preceding lemma, for positive integer m,

(a-b) \mid (a^m-b^m)

where \mid means "evenly divides". Substituting a = 2r, b = − 1, and m = s,

(2^r+1) \mid (2^{rs}+1),

and thus

(2^r+1) \mid (2^n+1).

Because 2r + 1 > 1, 2n + 1 is not prime when n is a positive integer that is not a power of 2.

Theorem: Any prime divisor of Fn = 2^{2^{\overset{n}{}}}+1 is of the form k2n + 2 + 1 whenever n is greater than one.

sketch of proof:

Euler knew that 2^{2^{ \overset {n}{}}} was Fn - 1 and thus congruent to -1(mod Fn), so using Fermat's little theorem he saw that any prime divisor p of Fn was such that 2n divided (p-1)/2. Therefore 2n + 1 divided p-1, so p was of form k2n + 1 + 1. Then, Edouard Lucas refined this, perhaps by noticing that x = 2^{3*2^{\overset{n-2}{}}}-2^{2^{\overset{n-2}{}}} is a square root of 2(mod Fn) for n greater than one. (Example: For n=3, F3=257 and 2^{3*(2^{3-2})}-2^{2^{3-2}} is 60, and 60 squared minus 2 is 3598=14*257.) Therefore, x^{2^{\overset{n+2}{}}} is congruent to 1(mod Fn) and so 2n + 2 divides p-1.

some details:

[2^{3*2^{n-2}} - 2^{2^{n-2}}] squared minus 2 is 2 times [2^{3*(2^{n-1})-1}-2^{2^{n}}+2^{2^{n-1}-1}-1], which can be factored as (2^{2^{n-1}-1}-1)(2^{2^{n}}+1).

[edit] Relationship to constructible polygons

An n-sided regular polygon can be constructed with compass and straightedge 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 = 2kp1p2...ps, where k is a nonnegative integer and the pi 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.

[edit] Applications of Fermat numbers

...Fermat number transform (Henri Nussbaumer)...

[edit] 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 relatively prime to P. Then take the result MOD P. The result is the new value for the RNG.

V_{j+1} = \left( A \times V_j \right) \bmod P (see Linear congruential generator, RANDU)

This is useful in computer science since most data structures have members with 2^X possible values. For example, a byte has 256 (2^8) 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.

[edit] Other interesting facts

A Fermat number cannot be a perfect number.(Luca 2000)

The series of reciprocals of all prime divisors of Fermat numbers is convergent.(Krizek, Luca, Somer 2002)

If nn + 1 is prime, there exists an integer m such that n = 22m. The equation nn + 1 = F2m+m holds at that time.

Let the largest prime factor of Fermat number Fn be P(Fn). Then,

P(F_n )\ge 2^{m+2}(4m+9)+1.

(Grytczuk, Luca and Wojtowicz, 2001）


[edit] Generalized Fermat numbers

Numbers of the form a2^n+b2^n, where a>1 are called as a generalized Fermat number. An odd prime p is a generalized Fermat prime if and only if there exists an integer i with i2=-1 (mod p) and i2<p.

[edit] References

  • 17 Lectures on Fermat Numbers: From Number Theory to Geometry, Michal Křížek, Florian Luca, Lawrence Somer, Springer, CMS Books 9, ISBN 0-387-95332-9 (This book contains an extensive list of references.)
  • S. W. Golomb, On the sum of the reciprocals of the Fermat numbers and related irrationalities, Canad. J. Math. 15(1963), 475--478.
  • Florian Luca, The anti-social Fermat number, Amer. Math. Monthly 107(2000), 171--173.
  • Michal Krizek, Florian Luca and Lawrence Somer(2002), On the convergence of series of reciprocals of primes related to the Fermat numbers, J. Number Theory 97(2002), 95--112.
  • A. Grytczuk, F. Luca and M. Wojtowicz(2001), Another note on the greatest prime factors of Fermat numbers, Southeast Asian Bull. Math. 25(2001), 111--115.

[edit] See also

[edit] External links

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