Fermat numbers

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

:$F_ = 2^ + 1,$

where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... .

If 2^''k'' + 1 is prime and ''k'' > 0, then ''k'' must be a power of 2, so 2^''k'' + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are ''F''₀ = 3, ''F''₁ = 5, ''F''₂ = 17, ''F''₃ = 257, and ''F''₄ = 65537 ; heuristics suggest that there are no more.

Basic properties

The Fermat numbers satisfy the following recurrence relations:

:$F_ = (F_-1)^+1$

:$F_ = F_ \cdots F_ + 2$

for ''n'' ≥ 1,

:$F_ = F_ + 2^F_ \cdots F_$

:$F_ = F_^2 - 2(F_-1)^2$

for ''n'' ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ ''i'' < ''j'' and ''F''_''i'' and ''F''_''j'' have a common factor ''a'' > 1. Then ''a'' divides both

:$F_ \cdots F_$

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 is an infinite sequence of distinct primes.

Further properties

* No Fermat prime can be expressed as the difference of two ''p''th powers, where ''p'' is an odd prime.
* With the exception of ''F''₀ and ''F''₁, the last digit of a Fermat number is 7.
* The sum of the reciprocals of all the Fermat numbers is irrational. (Solomon W. Golomb, 1963)

Primality

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. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed that

:$F_ = 2^ + 1 = 2^ + 1 = 4294967297 = 641 \times 6700417.$

Euler proved that every factor of ''F''_''n'' must have the form ''k''2^''n''+1 + 1 (later improved to ''k''2^''n''+2 + 1 by Lucas) for ''n'' ≥ 2.

That 641 is a factor of ''F''₅ can be deduced from the equalities 641 = 2⁷ Ã— 5 + 1 and 641 = 2⁴ + 5⁴. It follows from the first equality that 2⁷ Ã— 5 â‰¡ âˆ’1 (mod 641) and therefore (raising to the fourth power) that 2²⁸ Ã— 5⁴ â‰¡ 1 (mod 641). On the other hand, the second equality implies that 5⁴ â‰¡ âˆ’2⁴ (mod 641). These congruences imply that 2³² â‰¡ âˆ’1 (mod 641).

Fermat was probably aware of the form of the factors later proved by Euler, so it seems curious that he failed to follow through on the straightforward calculation to find the factor. One common explanation is that Fermat made a computational mistake.

There are no other known Fermat primes ''F''_''n'' with ''n'' > 4, but little is known about Fermat numbers for large ''n''. In fact, each of the following is an open problem:
* Is ''F''_''n'' composite for all ''n'' > 4?
* Are there infinitely many Fermat primes? ( Eisenstein 1844)
* Are there infinitely many composite Fermat numbers?
* Does a Fermat number exist that is not square-free?

, it is known that ''F''_''n'' is composite for , although of these, complete factorizations of ''F''_''n'' are known only for , and there are no known prime factors for and . The largest Fermat number known to be composite is ''F''_18233954, and its prime factor was discovered in October 2020.

Heuristic arguments

Heuristics suggest that ''F''₄ is the last Fermat prime.

The prime number theorem implies that a random integer in a suitable interval around ''N'' is prime with probability 1/ln ''N''. If one uses the heuristic that a Fermat number is prime with the same probability as a random integer of its size, and that ''F''₅, ..., ''F''₃₂ are composite, then the expected number of Fermat primes beyond ''F''₄ (or equivalently, beyond ''F''₃₂) should be
:$\sum_ \frac < \frac \sum_ \frac = \frac 2^ < 3.36 \times 10^.$
One may interpret this number as an upper bound for the probability that a Fermat prime beyond ''F''₄ exists.

This argument is not a rigorous proof. For one thing, it assumes that Fermat numbers behave "randomly", but the factors of Fermat numbers have special properties. Boklan and Conway published a more precise analysis suggesting that the probability that there is another Fermat prime is less than one in a billion.

Equivalent conditions

Let $F_n=2^+1$ be the ''n''th Fermat number. Pépin's test states that for ''n'' > 0,

:$F_n$ is prime if and only if $3^\equiv-1\pmod.$

The expression $3^$ can be evaluated modulo $F_n$ by repeated squaring. This makes the test a fast polynomial-time algorithm. But Fermat numbers grow so rapidly that only a handful of them can be tested in a reasonable amount of time and space.

There are some tests for numbers of the form ''k''2^''m'' + 1, such as factors of Fermat numbers, for primality.

: Proth's theorem (1878). Let $N$ = $k$$2$^$m$ + $1$ with odd $k$ < $2$^$m$. If there is an integer ''$a$'' such that
:: $a^ \equiv -1\pmod$
:then $N$ is prime. Conversely, if the above congruence does not hold, and in addition
:: $\left(\frac\right)=-1$ (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 some Fermat numbers, neither test gives a specific nontrivial factor. In fact, no specific prime factors are known for ''n'' = 20 and 24.

Factorization

Because of Fermat numbers' size, it is difficult to factorize or even to check primality. Pépin's test gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The elliptic curve method is a fast method for finding small prime divisors of numbers. Distributed computing project ''Fermatsearch'' has found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving Euler's above-mentioned result, proved in 1878 that every factor of the Fermat number $F_n$, with ''n'' at least 2, is of the form $k\times2^+1$ (see Proth number), where ''k'' is a positive integer. By itself, this makes it easy to prove the primality of the known Fermat primes.

Factorizations of the first twelve Fermat numbers are:

:

, only ''F''₀ to ''F''₁₁ have been completely factored. The distributed computing project Fermat Search is searching for new factors of Fermat numbers. The set of all Fermat factors is A050922 (or, sorted, A023394) in OEIS.

The following factors of Fermat numbers were known before 1950 (since then, digital computers have helped find more factors):

, 356 prime factors of Fermat numbers are known, and 312 Fermat numbers are known to be composite. Several new Fermat factors are found each year.

Pseudoprimes and Fermat numbers

Like composite numbers of the form 2^''p'' − 1, every composite Fermat number is a strong pseudoprime to base 2. This is because all strong pseudoprimes to base 2 are also Fermat pseudoprimes - i.e.

:$2^ \equiv 1 \pmod$

for all Fermat numbers.

In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers $F_ F_ \dots F_,$ $a > b > \dots > s > 1$ will be a Fermat pseudoprime to base 2 if and only if $2^s > a$.

Other theorems about Fermat numbers

A Fermat number cannot be a perfect number or part of a pair of amicable numbers.

The series of reciprocals of all prime divisors of Fermat numbers is convergent.

If ''n''^''n'' + 1 is prime, there exists an integer ''m'' such that ''n'' = 2^2''m''. The equation
''n''^''n'' + 1 = ''F''_{(2^''m''+''m'')}
holds in that case.

Let the largest prime factor of the Fermat number ''F''_''n'' be ''P''(''F''_''n''). Then,
:$P(F_n) \ge 2^(4n+9) + 1.$

Relationship to constructible polygons

Carl Friedrich Gauss developed the theory of Gaussian periods in his '' Disquisitiones Arithmeticae'' and formulated a sufficient condition for the constructibility of regular polygons. Gauss stated that this condition was also necessary, but never published a proof. Pierre Wantzel gave a full proof of necessity in 1837. The result is known as the Gauss–Wantzel theorem:

: An ''n''-sided regular polygon can be constructed with compass and straightedge if and only if ''n'' is 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, s'' are nonnegative integers and the ''p''_''i'' are distinct Fermat primes.

A positive integer ''n'' is of the above form if and only if its totient φ(''n'') is a power of 2.

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 is a primitive root modulo ''P'' (i.e., it is not a quadratic residue). Then take the result modulo ''P''. The result is the new value for the RNG.
: $V_ = (A \times V_j) \bmod P$ (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 that 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.

Generalized Fermat numbers

Numbers of the form $a^ \!\!+ b^$ with ''a'', ''b'' any coprime integers, ''a'' > ''b'' > 0, are called generalized Fermat numbers. An odd prime ''p'' is a generalized Fermat number if and only if ''p'' is congruent to 1 (mod 4). (Here we consider only the case ''n'' > 0, so 3 = $2^ \!+ 1$ is not a counterexample.)

An example of a probable prime of this form is 1215¹³¹⁰⁷² + 242¹³¹⁰⁷² (found by Kellen Shenton).

By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form $a^ \!\!+ 1$ as ''F''_''n''(''a''). In this notation, for instance, the number 100,000,001 would be written as ''F''₃(10). In the following we shall restrict ourselves to primes of this form, $a^ \!\!+ 1$, such primes are called "Fermat primes base ''a''". Of course, these primes exist only if ''a'' is even.

If we require ''n'' > 0, then Landau's fourth problem asks if there are infinitely many generalized Fermat primes ''F_n''(''a'').

Generalized Fermat primes

Because of the ease of proving their primality, generalized Fermat primes have become in recent years a topic for research within the field of number theory. Many of the largest known primes today are generalized Fermat primes.

Generalized Fermat numbers can be prime only for even , because if is odd then every generalized Fermat number will be divisible by 2. The smallest prime number $F_n(a)$ with $n>4$ is $F_5(30)$, or 30³² + 1. Besides, we can define "half generalized Fermat numbers" for an odd base, a half generalized Fermat number to base ''a'' (for odd ''a'') is $\frac$, and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base.

(In the list, the generalized Fermat numbers ($F_n(a)$) to an even are $a^ \!+ 1$, for odd , they are $\frac$. If is a perfect power with an odd exponent , then all generalized Fermat number can be algebraic factored, so they cannot be prime)

(For the smallest number $n$ such that $F_n(a)$ is prime, see )

(See for more information (even bases up to 1000), also see for odd bases)

(For the smallest prime of the form $F_n(a,b)$ (for odd $a+b$), see also )

(For the smallest even base such that $F_n(a)$ is prime, see )

The smallest base ''b'' such that ''b''^2''n'' + 1 is prime are
:2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444, ...

The smallest ''k'' such that (2''n'')^''k'' + 1 is prime are
:1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 4, 1, ... (The next term is unknown) (also see and )

A more elaborate theory can be used to predict the number of bases for which $F_n(a)$ will be prime for fixed $n$. The number of generalized Fermat primes can be roughly expected to halve as $n$ is increased by 1.

Largest known generalized Fermat primes

The following is a list of the 5 largest known generalized Fermat primes. The whole top-5 is discovered by participants in the PrimeGrid project.

On the Prime Pages one can find th
current top 100 generalized Fermat primes

See also

* Constructible polygon: which regular polygons are constructible partially depends on Fermat primes.
* Double exponential function
* Lucas' theorem
* Mersenne prime
* Pierpont prime
* Primality test
* Proth's theorem
* Pseudoprime
* Sierpiński number
* Sylvester's sequence

Notes

References

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* - This book contains an extensive list of references.
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External links

* Chris Caldwell
The Prime Glossary: Fermat number
at The Prime Pages.
* Luigi Morelli
History of Fermat Numbers
* John Cosgrave

* Wilfrid Keller

*
*
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* Yves Gallot

* Mark S. Manasse
Complete factorization of the ninth Fermat number
(original announcement)
* Peyton Hayslette
Largest Known Generalized Fermat Prime Announcement

{{Pierre de Fermat

Constructible polygons
Articles containing proofs
Unsolved problems in number theory
Large integers
Classes of prime numbers
Integer sequences