Fermat number

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^{\overset {n}{}}}+1

where n is a nonnegative integer. The first nine Fermat numbers are (sequence A000215 in the 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

F₈ = 2²⁵⁶ + 1 = 115792089237316195423570985008687907853269984665640564039457584007913129639937 = 1238926361552897 × 93461639715357977769163558199606896584051237541638188580280321

As of 2007, only the first 12 Fermat numbers have been completely factored. 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 ≡ (2^a)^b + 1 ≡ (−1)^b + 1 ≡ 0 (mod 2^a + 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.

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 F_i and F_j have a common factor a > 1. Then a divides both

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

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.

Further properties:

The number of digits D(n,b) of F_n expressed in the base b is

D(n,b)=\lfloor \log _{b}\left(2^{2^{\overset {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.

The sum of the reciprocals of all the Fermat numbers is irrational. (Solomon W. Golomb, 1963)

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 F_n 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 F_n with n > 4. However, very little is known about Fermat numbers with large n. [1] 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?

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. [2]

As of 2006 it is known that F_n is composite for 5 ≤ n ≤ 32, although complete factorizations of F_n 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 F_2478782, and its prime factor 3×2^2478785 + 1 was discovered by John B. 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 F_n.

Proth's theorem -- (1878) Let N = k2^m + 1 with odd k < 2^m. 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 = 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.

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 F_n > 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

F_{n}=x^{2}+y^{2}

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

.

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. Edouard Lucas proved in 1878 that every factor of Fermat number $F_{n}$ is of the form $2^{n+2}k+1$ , where k is a positive integer.

Original announcement of the factorization of the ninth Fermat number

Fermat's little theorem and pseudoprimes

Fermat's little theorem

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

Other theorems about Fermat numbers

Lemma: If n is a positive integer,

a^{n}-b^{n}=(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}.

proof:

(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}

=\sum _{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum _{k=0}^{n-1}a^{k}b^{n-k}

=a^{n}+\sum _{k=1}^{n-1}a^{k}b^{n-k}-\sum _{k=1}^{n-1}a^{k}b^{n-k}-b^{n}

=a^{n}-b^{n}

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

proof:

For $n=0$ , $2^{0}+1$ equals prime number 2. (This is why some sources count 2 as a sixth Fermat prime.)

If $n$ is a positive integer but not a power of 2, then $n=rs$ where $1\leq r<n$ , $1<s\leq n$ 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 $a=2^{r}$ , $b=-1$ , and $m=s$ ,

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

and thus

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

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

A theorem of Édouard Lucas: Any prime divisor p of F_n = $2^{2^{\overset {n}{}}}+1$ is of the form $k2^{n+2}+1$ whenever n is greater than one.

sketch of proof:

Let G_p 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 $2^{n+1}$ in G_p, so that, by Lagrange's theorem, p-1 is divisible by $2^{n+1}$ and p has the form $k2^{n+1}+1$ 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 in integer a such that a² -2 is divisible by p. Then the image of a has order $2^{n+2}$ in the group G_p and (using Lagrange's theorem again), p-1 is divisible by $2^{n+2}$ and p has the form $s2^{n+2}+1$ for some integer s.

In fact, it can be seen directly that 2 is a quadratic residue (mod p), since $(1+2^{2^{n-1}})^{2}\equiv 2^{1+2^{n-1}}$ (mod p). Since an odd power of 2 is a quadratic residue (mod p), so is 2 itself.

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 = 2^kp₁p₂...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

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 (2⁸) 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.(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 = 2²^{^m}. The equation nⁿ + 1 = F₍₂_^m_+m) holds at that time. [3]

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

P(F_{n})\geq 2^{m+2}(4m+9)+1

.

(Grytczuk, Luca and Wojtowicz, 2001）

Generalized Fermat numbers

Numbers of the form a^{2^n}+b^{2^n}, where a>1 are called as a generalized Fermat number. An odd prime p is a generalized Fermat number if and only if p is congruent to 1 ( mod 4).

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.
Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; sections A3,A12,B21.
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.

External links