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^n} + 1

where n is a nonnegative integer. The first eight Fermat numbers are :

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

Only the first 12 Fermat numbers have been completely factorised. These factorisations 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 2. (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.

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

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.

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 had 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. It did not take Euler very long to find 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. 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. ^*

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 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 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

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

Factorisation 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 $F_n$ 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's primes

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

=a^n-b^n

If $2^n+1$ is prime, then $n$ is a power of 2.

proof

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

If $n$ is a power of 2, or $n=ab$ where $1 < a, b < n$ and $b$ is odd.

2^{ab}+1=(2^a+1)\sum_{k=0}^{b-1} (2^a)^k(-1)^{b-1-k}.

Therefore, $2^a+1$ would divide $2^n+1$ , or $2^n+1$ is not prime.

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

...Fermat number transform...random number generation...

Other interesting facts

...F_n cannot be a perfect power, perfect, or part of amicable pair, etc...

Generalised Fermat numbers

...brief definition of L(p, m) and G(p, m) ...

References

17 Lectures on Fermat Numbers: From Number Theory to Geometry, Michal Křížek, Florian Luca, Lawrence Somer, Springer, CMS Books 9, ISBN 0387953329 (This book contains an extensive list of references.)

External links

Number theory | Euclidean plane geometry

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