In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form: where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... .
If 2k + 1 is prime and , then k itself must be a power of 2,For any positive odd number , where . so is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are , , , , and .
Basic properties The Fermat numbers satisfy the following recurrence relations:
for n ≥ 1,
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 Fi and Fj have a common factor . Then a divides both
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
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.Chris Caldwell, "Prime Links++: special forms" at The Prime Pages. 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 . The largest Fermat number known to be composite is F18233954, and its prime factor was discovered in October 2020.
Heuristic arguments Heuristics suggest that F4 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 F5, ..., F32 are composite, then the expected number of Fermat primes beyond F4 (or equivalently, beyond F32) should be
One may interpret this number as an upper bound for the probability that a Fermat prime beyond F4 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.
Anders Bjorn and Hans Riesel estimated the number of square factors of Fermat numbers from F5 onward as
in other words, there are unlikely to be any non-squarefree Fermat numbers, and in general square factors of are very rare for large n.
Equivalent conditions
Let be the nth Fermat number. Pépin's test states that for ,
is prime if and only if
The expression can be evaluated modulo 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
then is prime. Conversely, if the above congruence does not hold, and in addition (See Jacobi symbol) then 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 , with n at least 2, is of the form (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:
|- valign="top" |F0 ||=|| 21||+||1 ||=||3 is prime|| |- valign="top" |F1 ||=|| 22||+||1 ||=||5 is prime|| |- valign="top" |F2 ||=|| 24||+||1 ||=||17 is prime|| |- valign="top" |F3 ||=|| 28||+||1 ||=||257 is prime|| |- valign="top" |F4 ||=|| 216||+||1 ||=||65,537 is the largest known Fermat prime|| |- valign="top" |F5 ||=|| 232||+||1 ||=||4,294,967,297|| |- style="background:transparent; color:#B00000" | || || || || ||=||641 × 6,700,417 (fully factored 1732) |- valign="top" |F6 ||=|| 264||+||1 ||=||18,446,744,073,709,551,617 (20 digits) || |- style="background:transparent; color:#B00000" | || || || || ||=||274,177 × 67,280,421,310,721 (14 digits) (fully factored 1855) |- valign="top" |F7 ||=|| 2128||+||1 ||=||340,282,366,920,938,463,463,374,607,431,768,211,457 (39 digits)|| |- style="background:transparent; color:#B00000" | || || || || ||=||59,649,589,127,497,217 (17 digits) × 5,704,689,200,685,129,054,721 (22 digits) (fully factored 1970) |- valign="top" |F8 ||=|| 2256||+||1 ||=||115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,937 (78 digits)|| |- style="background:transparent; color:#B00000; vertical-align:top" | || || || || ||=||1,238,926,361,552,897 (16 digits) × 93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321 (62 digits) (fully factored 1980) |- valign="top" |F9 ||=|| 2512||+||1 ||=||13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,097 (155 digits)|| |- style="background:transparent; color:#B00000; vertical-align:top" | || || || || ||=|| 2,424,833 × 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 (49 digits) × 741,640,062,627,530,801,524,787,141,901,937,474,059,940,781,097,519,023,905,821,316,144,415,759,504,705,008,092,818,711,693,940,737 (99 digits) (fully factored 1990) |- valign="top" |F10 ||=|| 21024||+||1 ||=||179,769,313,486,231,590,772,930...304,835,356,329,624,224,137,217 (309 digits)|| |- style="background:transparent; color:#B00000; vertical-align:top" | || || || || ||=||45,592,577 × 6,487,031,809 × 4,659,775,785,220,018,543,264,560,743,076,778,192,897 (40 digits) × 130,439,874,405,488,189,727,484...806,217,820,753,127,014,424,577 (252 digits) (fully factored 1995) |- valign="top" |F11 ||=|| 22048||+||1 ||=||32,317,006,071,311,007,300,714,8...193,555,853,611,059,596,230,657 (617 digits)|| |- style="background:transparent; color:#B00000; vertical-align:top" | || || || || ||=|| 319,489 × 974,849 × 167,988,556,341,760,475,137 (21 digits) × 3,560,841,906,445,833,920,513 (22 digits) × 173,462,447,179,147,555,430,258...491,382,441,723,306,598,834,177 (564 digits) (fully factored 1988)
, only F0 to F11 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):
Year Finder Fermat number Factor 1732 Euler 1732 Euler (fully factored) 1855 Clausen 1855 Clausen (fully factored) 1877 Pervushin 1878 Pervushin 1886 Seelhoff 1899 Cunningham 1899 Cunningham 1903 Western 1903 Western 1903 Western 1903 Western 1903 Cullen 1906 Morehead 1925 Kraitchik
, 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.,
for all Fermat numbers.
In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers will be a Fermat pseudoprime to base 2 if and only if .
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 , there exists an integer m such that . The equation
holds in that case.Jeppe Stig Nielsen, "S(n) = n^n + 1".
Let the largest prime factor of the Fermat number Fn be P(Fn). Then,
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. (see linear congruential generator)