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 = 2²ⁿ+1

where n is a non-negative integer. The first eight Fermat numbers are (see 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

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 2ⁿ+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 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

[math]\displaystyle{ F_{n} = (F_{n-1}-1)^{2}+1 }[/math]

[math]\displaystyle{ F_{n} = F_{n-1} + 2^{2^{n-1}}F_{0} \cdots F_{n-2} }[/math]

[math]\displaystyle{ F_{n} = F_{n-1}^2 - 2(F_{n-2}-1)^2 }[/math]

[math]\displaystyle{ F_{n} = F_{0} \cdots F_{n-1} + 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]\displaystyle{ F_{0} \cdots F_{j-1} }[/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]\displaystyle{ 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 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

[math]\displaystyle{ 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 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 isn't hard to check this, but there is an interesting proof for this: [math]\displaystyle{ 641=2^4+5^4=5*2^7+1 }[/math] using this we can get

[math]\displaystyle{ 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 F₅ 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 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?
• Is every Fermat number 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]\displaystyle{ 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 (John Cosgrave: "Fermat 6").

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 Fermat 2^218233954+1, and its prime factor Proth 7•2^18233956+1 was discovered by Ryan Propper on 2020-10-01.

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

• Proth's theorem -- (1878) Let N = k•2^m+1 with odd k < 2^m. If there is an integer a such that

[math]\displaystyle{ a^{(N-1)/2} \equiv -1 \,\pmod N }[/math]

then N is prime. Conversely, if the above congruence does not hold, and in addition

[math]\displaystyle{ \left(\frac{a}{N}\right)=-1 }[/math] (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.

• Theorem: Let n>0 then [math]\displaystyle{ F_n }[/math] is prime if and only if it divides [math]\displaystyle{ S_{2^n-2} }[/math] where [math]\displaystyle{ S_0=5 }[/math] and [math]\displaystyle{ S_{k+1}=S^2_{k}-2 }[/math] (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]\displaystyle{ F_{n}=\left(2^{2^{n-1}}\right)^{2}+1^{2} }[/math]

When [math]\displaystyle{ F_{n} = x^2 + y^2 }[/math] not of the form shown above, a proper factor is:

[math]\displaystyle{ \gcd(x + 2^{2^{n-1}} y, F_{n}) }[/math]

Example 1: F₅ = 62264² + 20449², so a proper factor is [math]\displaystyle{ \gcd(62264\, +\, 2^{2^4}\, 20449,\, F_{5}) = 641 }[/math].

Example 2: F₆ = 4046803256² + 1438793759², so a proper factor is [math]\displaystyle{ \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]\displaystyle{ F_n }[/math] is of the form [math]\displaystyle{ 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 n-sided 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^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...

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.)

Number classes

General numbers

Integer Natural Even Odd Rational Irrational Real Prime Probable prime

Special numbers

Mersenne 2n-1 Fermat 22n+1 Proth k•2n+1 Riesel k•2n-1 Woodall n•2n-1 Cullen n•2n+1 Double Mersenne 22p-1-1 Perfect 2n-1(2n-1) Leyland xy ± yx Gen.Repunit (b^n-1)/(b-1)

Prime numbers

Mersenne Proth Riesel Williams (b±1)bn±1 Carol/Kynea (bn±1)2-2 Multifactorial n!j±1