Fermat number

The Fermat numbers satisfy the following recurrence relations: : F_{n} = (F_{n-1}-1)^{2}+1 : F_{n} = F_{0} F_{1} \cdots F_{n-1} + 2 for n ≥ 1, : F_{n} = F_{n-1} + 2^{2^{n-1}}F_{0} F_{1} \cdots F_{n-2} : F_{n} = F_{n-1}^2 - 2(F_{n-2}-1)^2 for . 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 and F''i and Fj have a common factor . Then a divides both :F_{0} F_{1} \cdots F_{j-1} and Fj; hence a divides their difference, 2. Since , this forces . 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 is an infinite sequence of distinct primes. Further properties • No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime. • With the exception of F0 and F1, the last decimal 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 F0, ..., F4 are easily shown to be prime. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed, by dividing by 641 that : F_{5} = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297 = 641 \times 6700417. Euler proved that every factor of Fn must have the form (later improved to by Lucas) for . That 641 is a factor of F5 can be deduced, in hindsight, as follows: From the equalities 641 = 27 × 5 + 1 and 641 = 24 + 54. It follows from the first equality that 27 × 5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 228 × 54 ≡ 1 (mod 641). On the other hand, the second equality implies that 54 ≡ −24 (mod 641). These congruences imply that 232 ≡ −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 Fn with , but little is known about Fermat numbers for large n. In fact, each of the following is an open problem: • Is Fn composite for all ? • 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 Fn is composite for , although of these, complete factorizations of Fn are known only for , and there are no known prime factors for and . Anders Bjorn and Hans Riesel estimated the number of square factors of Fermat numbers from F5 onward as : \sum_{n \ge 5} \sum_{k \ge 1} \frac{1}{k (k 2^n + 1) \ln(k 2^n)} in other words, there are unlikely to be any non-squarefree Fermat numbers, and in general square factors of a^{2^n} + b^{2^n} are very rare for large n. Equivalent conditions Let F_n=2^{2^n}+1 be the nth Fermat number. Pépin's test states that for , :F_n is prime if and only if 3^{(F_n-1)/2}\equiv-1\pmod{F_n}. The expression 3^{(F_n-1)/2} 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 , such as factors of Fermat numbers, for primality. :'''Proth's theorem' (1878). Let with odd . If there is an integer a'' such that :: a^{(N-1)/2} \equiv -1\pmod{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 , then the above Jacobi symbol is always equal to −1 for , 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 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 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^{n+2}+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 12 Fermat numbers are: : , only F0 to F11 have been completely factored. 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): , 375 prime factors of Fermat numbers are known, and 330 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 2p − 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^{F_n-1} \equiv 1 \pmod{F_n} for all Fermat numbers. In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers F_{a} F_{b} \dots F_{s}, 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 is prime and n \ge 2, there exists an integer m such that . The equation holds in that case. Let the largest prime factor of the Fermat number Fn be P(Fn). Then, :P(F_n) \ge 2^{n+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 , where k, s are nonnegative integers and the pi 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 , 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_{j+1} = (A \times V_j) \bmod P (see linear congruential generator) This is useful in computer science, since most data structures have members with 2X 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 repetitions, the sequence repeats. A poorly chosen multiplier can result in the sequence repeating sooner than . ==Generalized Fermat numbers==

Numbers of the form \frac{a^{2^n}+b^{2^n}}{gcd(a+b,2)} with a, b any coprime integers, , 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 , so {{nowrap|1=3 = 2^{2^{0}} \!+ 1}} is not a counterexample.) An example of a probable prime of this form is 200262144 + 119262144 (found by Kellen Shenton). By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form a^{2^{ \overset{n} {}}} \!\!+ 1 as Fn(a). In this notation, for instance, the number 100,000,001 would be written as F3(10). In the following we shall restrict ourselves to primes of this form, a^{2^{ \overset{n} {}}} \!\!+ 1, such primes are called "Fermat primes base a". Of course, these primes exist only if a is even. Generalized Fermat primes of the form Fn(a) 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{a^{2^n} \!+ 1}{2}, and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base. In this list, the generalized Fermat numbers (F_n(a)) to an even are a^{2^n} \!+ 1, for odd , they are \frac{a^{2^n} \!\!+ 1}{2}. If is a perfect power with an odd exponent , then all generalized Fermat number can be algebraic factored, so they cannot be prime. See for even bases up to 1000, and for odd bases. For the smallest number n such that F_n(a) is prime, see . For the smallest even base such that F_n(a) is prime, see . The generalized Fermat prime F14(71) is the largest known generalized Fermat prime in bases b ≤ 1000, it is proven prime by elliptic curve primality proving. The smallest even base b such that Fn(b) = b2n + 1 (for given n = 0, 1, 2, ...) is prime are :2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444, 2524190, ... The smallest odd base b such that Fn(b) = (b2n + 1)/2 (for given n = 0, 1, 2, ...) is prime (or probable prime) are :3, 3, 3, 9, 3, 3, 3, 113, 331, 513, 827, 799, 3291, 5041, 71, 220221, 23891, 11559, 187503, 35963, ... Conversely, the smallest k such that (2n)k + 1 (for given n) 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. Generalized Fermat primes of the form Fn(a, b) It is also possible to construct generalized Fermat primes of the form a^{2^n} + b^{2^n}. As in the case where b=1, numbers of this form will always be divisible by 2 if a+b is even, but it is still possible to define generalized half-Fermat primes of this type. For the smallest prime of the form F_n(a,b) (for odd a+b), see also . Largest known generalized Fermat primes The following is a list of the ten largest known generalized Fermat primes. The whole top-10 is discovered by participants in the PrimeGrid project. On the Prime Pages one can find the current top 20 generalized Fermat primes and the current top 100 generalized Fermat primes. ==See also==