Fermat number
Named after | Pierre de Fermat |
---|---|
Number of known terms | 5 |
Conjectured number of terms | 5 |
Subsequence of | Fermat numbers |
First terms | 3, 5, 17, 257, 65537 |
Largest known term | 65537 |
OEIS index | A019434 |
In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form
where n is a nonnegative integer. The first few Fermat numbers are:
If 2^{k} + 1 is prime, and k > 0, it can be shown that k must be a power of two. (If k = ab where 1 ≤ a, b ≤ k and b is odd, then 2^{k} + 1 = (2^{a})^{b} + 1 ≡ (−1)^{b} + 1 = 0 (mod 2^{a} + 1). See below for a complete proof.) In other words, every prime of the form 2^{k} + 1 (other than 2 = 2^{0} + 1) is a Fermat number, and such primes are called Fermat primes. As of 2016, the only known Fermat primes are F_{0}, F_{1}, F_{2}, F_{3}, and F_{4} (sequence A019434 in the OEIS).
Basic properties
The Fermat numbers satisfy the following recurrence relations:
for n ≥ 1,
for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last 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
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.
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 F_{0} and F_{1}, the last digit of a Fermat number is 7.
- The sum of the reciprocals of all the Fermat numbers (sequence A051158 in the OEIS) is irrational. (Solomon W. Golomb, 1963)
Primality of Fermat numbers
Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured (but admitted he could not prove) that all Fermat numbers are prime. Indeed, the first five Fermat numbers F_{0},...,F_{4} are easily shown to be prime. However, this conjecture was refuted by Leonhard Euler in 1732 when he showed that
Euler proved that every factor of F_{n} must have the form k2^{n+1} + 1 (later improved to k2^{n+2} + 1 by Lucas).
The fact that 641 is a factor of F_{5} can be easily deduced from the equalities 641 = 2^{7}×5+1 and 641 = 2^{4} + 5^{4}. It follows from the first equality that 2^{7}×5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 2^{28}×5^{4} ≡ 1 (mod 641). On the other hand, the second equality implies that 5^{4} ≡ −2^{4} (mod 641). These congruences imply that −2^{32} ≡ 1 (mod 641).
Fermat was probably aware of the form of the factors later proved by Euler, so it seems curious why he failed to follow through on the straightforward calculation to find the factor.^{[1]} One common explanation is that Fermat made a computational mistake.
There are no other known Fermat primes F_{n} with n > 4. However, little is known about Fermat numbers with large n.^{[2]} 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)^{[3]}
- Are there infinitely many composite Fermat numbers?
- Does a Fermat number exist that is not square-free?
As of 2014, it is known that F_{n} is composite for 5 ≤ n ≤ 32, although amongst these, complete factorizations of F_{n} are known only for 0 ≤ n ≤ 11, and there are no known prime factors for n = 20 and n = 24.^{[4]} The largest Fermat number known to be composite is F_{3329780}, and its prime factor 193 × 2^{3329782} + 1, a megaprime, was discovered by the PrimeGrid collaboration in July 2014.^{[4]}^{[5]}
Heuristic arguments for density
There are several probabilistic arguments for estimating the number of Fermat primes. However these arguments give quite different estimates, depending on how much information about Fermat numbers one uses, and some predict no further Fermat primes while others predict infinitely many Fermat primes.
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 about 1/ln(n). Therefore, the total expected number of Fermat primes is at most
This argument is not 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. For example, a similar argument applied to the numbers of the form 2^{n}+1 suggests that infinitely many of them should be prime, which would mean that there are an infinite number of Fermat primes (though much the same argument applied to the numbers 2^{n} suggests the ridiculous result that infinitely many of them should be prime, so such probability arguments should not be taken too seriously).
If (more sophisticatedly) we regard the conditional probability that n is prime, given that we know all its prime factors exceed B, as at most A ln(B) / ln(n), then using Euler's theorem that the least prime factor of F_{n} exceeds 2^{n + 1}, we would find instead
Although such arguments engender the belief that there are only finitely many Fermat primes, one can also produce arguments for the opposite conclusion. Suppose we regard the conditional probability that n is prime, given that we know all its prime factors are 1 modulo M, as at least CM/ln(n). Then using Euler's result that M = 2^{n + 1} we would find that the expected total number of Fermat primes was at least
and indeed this argument predicts that an asymptotically constant fraction of Fermat numbers are prime.
Equivalent conditions of primality
Let be the nth Fermat number. Pépin's test states that for n > 0,
- is prime if and only if
The expression can be evaluated modulo by repeated squaring. This makes the test a fast polynomial-time algorithm. However, Fermat numbers grow so rapidly that only a handful of Fermat numbers can be tested in a reasonable amount of time and space.
There some tests that can be used to test numbers of the form k2^{m} + 1, such as factors of Fermat numbers, for primality.
- Proth's theorem (1878)—Let N = k2^{m} + 1 with odd k < 2^{m}. If there is an integer a such that
- then N is prime. Conversely, if the above congruence does not hold, and in addition
- (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 of Fermat numbers
Because of the size of Fermat numbers, 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 successfully found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving the above-mentioned result by Euler, proved in 1878 that every factor of 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 twelve Fermat numbers are:
F_{0} | = | 2^{1} | + | 1 | = | 3 is prime | |
F_{1} | = | 2^{2} | + | 1 | = | 5 is prime | |
F_{2} | = | 2^{4} | + | 1 | = | 17 is prime | |
F_{3} | = | 2^{8} | + | 1 | = | 257 is prime | |
F_{4} | = | 2^{16} | + | 1 | = | 65,537 is the largest known Fermat prime | |
F_{5} | = | 2^{32} | + | 1 | = | 4,294,967,297 | |
= | 641 × 6,700,417 | ||||||
F_{6} | = | 2^{64} | + | 1 | = | 18,446,744,073,709,551,617 (20 digits) | |
= | 274,177 × 67,280,421,310,721 (14 digits) | ||||||
F_{7} | = | 2^{128} | + | 1 | = | 340,282,366,920,938,463,463,374,607,431,768,211,457 (39 digits) | |
= | 59,649,589,127,497,217 (17 digits) × 5,704,689,200,685,129,054,721 (22 digits) | ||||||
F_{8} | = | 2^{256} | + | 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) | |
= | 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) | ||||||
F_{9} | = | 2^{512} | + | 1 | = | 13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,0 30,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,6 49,006,084,097 (155 digits) | |
= | 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) | ||||||
F_{10} | = | 2^{1024} | + | 1 | = | 179,769,313,486,231,590,772,930...304,835,356,329,624,224,137,217 (309 digits) | |
= | 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) | ||||||
F_{11} | = | 2^{2048} | + | 1 | = | 32,317,006,071,311,007,300,714,8...193,555,853,611,059,596,230,657 (617 digits) | |
= | 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) | ||||||
As of 2016, only F_{0} to F_{11} have been completely factored.^{[4]} The distributed computing project Fermat Search is searching for new factors of Fermat numbers.^{[6]} The set of all Fermat factors is A050922 (or, sorted, A023394) in OEIS.
The following factors of Fermat numbers were known before 1950 (since the 1950s 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 | ||
As of September 2016, 331 prime factors of Fermat numbers are known, and 288 Fermat numbers are known to be composite.^{[4]}
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.
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 .^{[7]}
Selfridge's Conjecture
John L. Selfridge made an conjecture. Let g(n) be the number of distinct prime factors of 2^{2n} + 1 (sequence A046052 in the OEIS). Then g(n) is not monotonic (nondecreasing). If another Fermat prime exists, that would imply the conjecture.^{[8]}
Other theorems about Fermat numbers
Lemma: If n is a positive integer,
Proof:
Theorem: If is an odd prime, then is a power of 2.
Proof:
If is a positive integer but not a power of 2, must have an odd prime factor, .
This means that where .
By the preceding lemma, for positive integer ,
where means "evenly divides". Substituting , , and and using that is odd,
and thus
Because , it follows that is not prime. Therefore, by contraposition must be a power of 2.
Theorem: A Fermat prime cannot be a Wieferich prime.
Proof: We show if is a Fermat prime (and hence by the above, m is a power of 2), then the congruence does not hold.
It is easy to show . Now write . If the given congruence holds, then , and therefore
Hence ,and therefore . This leads to
, which is impossible since .
A theorem of Édouard Lucas: Any prime divisor p of F_{n} = is of the form whenever n is greater than one.
Sketch of proof:
Let G_{p} denote the group of non-zero elements of the integers (mod p) under multiplication, which has order p-1. Notice that 2 (strictly speaking, its image (mod p)) has multiplicative order dividing in G_{p} (since is the square of which is -1 mod F_{n}), so that, by Lagrange's theorem, p-1 is divisible by and p has the form for some integer k, as Euler knew. Édouard Lucas went further. Since n is greater than 1, the prime p above is congruent to 1 (mod 8). Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue (mod p), that is, there is integer a such that a^{2} -2 is divisible by p. Then the image of a has order in the group G_{p} and (using Lagrange's theorem again), p-1 is divisible by and p has the form for some integer s.
In fact, it can be seen directly that 2 is a quadratic residue (mod p), since (mod p). Since an odd power of 2 is a quadratic residue (mod p), so is 2 itself.
Relationship to constructible polygons
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 is a nonnegative integer 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.
This is useful in computer science since most data structures have members with 2^{X} possible values. For example, a byte has 256 (2^{8}) possible values (0–255). Therefore, to fill a byte or bytes with random values a random number generator which 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.
Other interesting facts
A Fermat number cannot be a perfect number or part of a pair of amicable numbers. (Luca 2000)
The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Křížek, Luca & Somer 2002)
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 at that time.^{[9]}
Let the largest prime factor of Fermat number F_{n} be P(F_{n}). Then,
Generalized Fermat numbers
Numbers of the form 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 = is not a counterexample.)
By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form 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, , 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 hot 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 with is , or 30^{32}+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 , 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 () to an even are , for odd , they are . If is a perfect power with an odd exponent (sequence A070265 in the OEIS), then all generalized Fermat number can be algrabic factored, so they cannot be prime)
(For the smallest number such that is prime, see A253242)
numbers such that is prime | numbers such that is prime | numbers such that is prime | numbers such that is prime | ||||
---|---|---|---|---|---|---|---|
2 | 0, 1, 2, 3, 4, ... | 18 | 0, ... | 34 | 2, ... | 50 | ... |
3 | 0, 1, 2, 4, 5, 6, ... | 19 | 1, ... | 35 | 1, 2, 6, ... | 51 | 1, 3, 6, ... |
4 | 0, 1, 2, 3, ... | 20 | 1, 2, ... | 36 | 0, 1, ... | 52 | 0, ... |
5 | 0, 1, 2, ... | 21 | 0, 2, 5, ... | 37 | 0, ... | 53 | 3, ... |
6 | 0, 1, 2, ... | 22 | 0, ... | 38 | ... | 54 | 1, 2, 5, ... |
7 | 2, ... | 23 | 2, ... | 39 | 1, 2, ... | 55 | ... |
8 | (none) | 24 | 1, 2, ... | 40 | 0, 1, ... | 56 | 1, 2, ... |
9 | 0, 1, 3, 4, 5, ... | 25 | 0, 1, ... | 41 | 4, ... | 57 | 0, 2, ... |
10 | 0, 1, ... | 26 | 1, ... | 42 | 0, ... | 58 | 0, ... |
11 | 1, 2, ... | 27 | (none) | 43 | 3, ... | 59 | 1, ... |
12 | 0, ... | 28 | 0, 2, ... | 44 | 4, ... | 60 | 0, ... |
13 | 0, 2, 3, ... | 29 | 1, 2, 4, ... | 45 | 0, 1, ... | 61 | 0, 1, 2, ... |
14 | 1, ... | 30 | 0, 5, ... | 46 | 0, 2, 9, ... | 62 | ... |
15 | 1, ... | 31 | ... | 47 | 3, ... | 63 | ... |
16 | 0, 1, 2, ... | 32 | (none) | 48 | 2, ... | 64 | (none) |
17 | 2, ... | 33 | 0, 3, ... | 49 | 1, ... | 65 | 1, 2, 5, ... |
(See ^{[10]}^{[11]} for more information (even bases up to 1000), also see ^{[12]} for odd bases)
(For the smallest prime of the form (for odd ), see also A111635)
numbers such that is prime | ||
---|---|---|
2 | 1 | 0, 1, 2, 3, 4, ... |
3 | 1 | 0, 1, 2, 4, 5, 6, ... |
3 | 2 | 0, 1, 2, ... |
4 | 1 | 0, 1, 2, 3, ... |
4 | 3 | 0, 2, 4, ... |
5 | 1 | 0, 1, 2, ... |
5 | 2 | 0, 1, 2, ... |
5 | 3 | 1, 2, 3, ... |
5 | 4 | 1, 2, ... |
6 | 1 | 0, 1, 2, ... |
6 | 5 | 0, 1, 3, 4, ... |
7 | 1 | 2, ... |
7 | 2 | 1, 2, ... |
7 | 3 | 0, 1, 8, ... |
7 | 4 | 0, 2, ... |
7 | 5 | 1, 4, ... |
7 | 6 | 0, 2, 4, ... |
8 | 1 | (none) |
8 | 3 | 0, 1, 2, ... |
8 | 5 | 0, 1, 2, ... |
8 | 7 | 1, 4, ... |
9 | 1 | 0, 1, 3, 4, 5, ... |
9 | 2 | 0, 2, ... |
9 | 4 | 0, 1, ... |
9 | 5 | 0, 1, 2, ... |
9 | 7 | 2, ... |
9 | 8 | 0, 2, 5, ... |
10 | 1 | 0, 1, ... |
10 | 3 | 0, 1, 3, ... |
10 | 7 | 0, 1, 2, ... |
10 | 9 | 0, 1, 2, ... |
11 | 1 | 1, 2, ... |
11 | 2 | 0, 2, ... |
11 | 3 | 0, 3, ... |
11 | 4 | 1, 2, ... |
11 | 5 | 1, ... |
11 | 6 | 0, 1, 2, ... |
11 | 7 | 2, 4, 5, ... |
11 | 8 | 0, 6, ... |
11 | 9 | 1, 2, ... |
11 | 10 | 5, ... |
12 | 1 | 0, ... |
12 | 5 | 0, 4, ... |
12 | 7 | 0, 1, 3, ... |
12 | 11 | 0, ... |
13 | 1 | 0, 2, 3, ... |
13 | 2 | 1, 3, 9, ... |
13 | 3 | 1, 2, ... |
13 | 4 | 0, 2, ... |
13 | 5 | 1, 2, 4, ... |
13 | 6 | 0, 6, ... |
13 | 7 | 1, ... |
13 | 8 | 1, 3, 4, ... |
13 | 9 | 0, 3, ... |
13 | 10 | 0, 1, 2, 4, ... |
13 | 11 | 2, ... |
13 | 12 | 1, 2, 5, ... |
14 | 1 | 1, ... |
14 | 3 | 0, 3, ... |
14 | 5 | 0, 2, 4, 8, ... |
14 | 9 | 0, 1, 8, ... |
14 | 11 | 1, ... |
14 | 13 | 2, ... |
15 | 1 | 1, ... |
15 | 2 | 0, 1, ... |
15 | 4 | 0, 1, ... |
15 | 7 | 0, 1, 2, ... |
15 | 8 | 0, 2, 3, ... |
15 | 11 | 0, 1, 2, ... |
15 | 13 | 1, 4, ... |
15 | 14 | 0, 1, 2, 4, ... |
16 | 1 | 0, 1, 2, ... |
16 | 3 | 0, 2, 8, ... |
16 | 5 | 1, 2, ... |
16 | 7 | 0, 6, ... |
16 | 9 | 1, 3, ... |
16 | 11 | 2, 4, ... |
16 | 13 | 0, 3, ... |
16 | 15 | 0, ... |
(For the smallest even base such that is prime, see A056993)
bases such that is prime (only consider even ) | OEIS sequence | |
---|---|---|
0 | 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, ... | A006093 |
1 | 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, ... | A005574 |
2 | 2, 4, 6, 16, 20, 24, 28, 34, 46, 48, 54, 56, 74, 80, 82, 88, 90, 106, 118, 132, 140, 142, 154, 160, 164, 174, 180, 194, 198, 204, 210, 220, 228, ... | A000068 |
3 | 2, 4, 118, 132, 140, 152, 208, 240, 242, 288, 290, 306, 378, 392, 426, 434, 442, 508, 510, 540, 542, 562, 596, 610, 664, 680, 682, 732, 782, ... | A006314 |
4 | 2, 44, 74, 76, 94, 156, 158, 176, 188, 198, 248, 288, 306, 318, 330, 348, 370, 382, 396, 452, 456, 470, 474, 476, 478, 560, 568, 598, 642, ... | A006313 |
5 | 30, 54, 96, 112, 114, 132, 156, 332, 342, 360, 376, 428, 430, 432, 448, 562, 588, 726, 738, 804, 850, 884, 1068, 1142, 1198, 1306, 1540, 1568, ... | A006315 |
6 | 102, 162, 274, 300, 412, 562, 592, 728, 1084, 1094, 1108, 1120, 1200, 1558, 1566, 1630, 1804, 1876, 2094, 2162, 2164, 2238, 2336, 2388, ... | A006316 |
7 | 120, 190, 234, 506, 532, 548, 960, 1738, 1786, 2884, 3000, 3420, 3476, 3658, 4258, 5788, 6080, 6562, 6750, 7692, 8296, 9108, 9356, 9582, ... | A056994 |
8 | 278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332, ... | A056995 |
9 | 46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992, ... | A057465 |
10 | 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, ... | A057002 |
11 | 150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654, ... | A088361 |
12 | 1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696, ... | A088362 |
13 | 30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894, 176206, 180924, 201170, 212724, 222764, 225174, 241600, ... | A226528 |
14 | 67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, 422666, 426690, 449732, 462470, 468144, 498904, 506664, ... | A226529 |
15 | 70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, 553602, 743788, 825324, 831648, 855124, 999236, 1041870, ... | A226530 |
16 | 48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540, ... | A251597 |
17 | 62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192, ... | A253854 |
18 | 24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, ... | A244150 |
19 | 75898, 341112, 356926, 475856, ... | A243959 |
The smallest base b such that b^{2n} + 1 is prime are
- 2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, ... (sequence A056993 in the OEIS)
The smallest k such that (2n)^{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) (sequence A079706 in the OEIS) (also see A228101 and A084712)
A more elaborate theory can be used to predict the number of bases for which will be prime for fixed . The number of generalized Fermat primes can be roughly expected to halve as is increased by 1.
Largest known generalized Fermat primes
The following is a list of the 10 largest known generalized Fermat primes.^{[13]} They are all megaprimes. As of April 2015 the whole top-10 was discovered by participants in the PrimeGrid project.
Rank | Prime rank^{[14]} | Prime number | Generalized Fermat notation | Number of digits | Found date | reference |
---|---|---|---|---|---|---|
1 | 16 | 475856^{524288} + 1 | F_{19}(475856) | 2,976,633 | 2012 August 8 | ^{[15]} |
2 | 17 | 356926^{524288} + 1 | F_{19}(356926) | 2,911,151 | 2012 June 20 | ^{[16]} |
3 | 18 | 341112^{524288} + 1 | F_{19}(341112) | 2,900,832 | 2012 June 15 | ^{[17]} |
4 | 21 | 75898^{524288} + 1 | F_{19}(75898) | 2,558,647 | 2011 November 19 | ^{[18]} |
5 | 42 | 773620^{262144} + 1 | F_{18}(773620) | 1,543,643 | 2012 April 19 | ^{[19]} |
6 | 45 | 676754^{262144} + 1 | F_{18}(676754) | 1,528,413 | 2012 February 12 | ^{[20]} |
7 | 48 | 525094^{262144} + 1 | F_{18}(525094) | 1,499,526 | 2012 January 18 | ^{[21]} |
8 | 52 | 361658^{262144} + 1 | F_{18}(361658) | 1,457,075 | 2011 October 29 | ^{[22]} |
9 | 62 | 145310^{262144} + 1 | F_{18}(145310) | 1,353,265 | 2011 February 8 | ^{[23]} |
10 | 74 | 40734^{262144} + 1 | F_{18}(40734) | 1,208,473 | 2011 March 8 | ^{[24]} |
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
- ↑ Křížek, Luca & Somer 2001, p. 38, Remark 4.15
- ↑ Chris Caldwell, "Prime Links++: special forms" at The Prime Pages.
- ↑ Ribenboim 1996, p. 88.
- 1 2 3 4 Keller, Wilfrid (February 7, 2012), "Prime Factors of Fermat Numbers", ProthSearch.net, retrieved September 14, 2016
- ↑ "PrimeGrid's Mega Prime Search – 193*2^3329782+1 (official announcement)" (PDF). PrimeGrid. Retrieved 7 August 2014.
- ↑ FermatSearch.org
- ↑ https://books.google.com/books?id=hgfSBwAAQBAJ&lpg=PA132&ots=On_KUoiNZp&dq=cipolla%20fermat%201904&pg=PA132#v=onepage&q&f=false
- ↑ Prime Numbers: A Computational Perspective, Richard Crandall and Carl Pomerance, Second edition, Springer, 2011 Look up Selfridge's Conjecture in the Index.
- ↑ Jeppe Stig Nielsen, "S(n) = n^n + 1".
- ↑ Generalized Fermat primes for bases up to 1000
- ↑ Generalized Fermat primes for bases up to 1030
- ↑ Generalized Fermat primes in odd bases
- ↑ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
- ↑ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 475856^524288+1" (PDF). Primegrid. Retrieved 21 August 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 356926^524288+1" (PDF). Primegrid. Retrieved 30 July 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 341112^524288+1" (PDF). Primegrid. Retrieved 9 July 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 75898^524288+1" (PDF). Primegrid. Retrieved 9 July 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 773620^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 676754^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 525094^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 361658^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 145310^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
- ↑ "PrimeGrid's Generalized Fermat Prime Search - 40734^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
References
- Golomb, S. W. (January 1, 1963), "On the sum of the reciprocals of the Fermat numbers and related irrationalities" (PDF), Canadian Journal of Mathematics, Canadian Mathematical Society, 15: 475–478, doi:10.4153/CJM-1963-051-0, ISSN 0008-414X
- Grytczuk, A.; Luca, F. & Wójtowicz, M. (2001), "Another note on the greatest prime factors of Fermat numbers", Southeast Asian Bulletin of Mathematics, Springer-Verlag, 25 (1): 111–115, doi:10.1007/s10012-001-0111-4, ISSN 0129-2021
- Guy, Richard K. (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics, 1 (3rd ed.), New York: Springer Verlag, pp. A3, A12, B21, ISBN 0-387-20860-7
- Křížek, Michal; Luca, Florian & Somer, Lawrence (2001), 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS books in mathematics, 10, New York: Springer, ISBN 0-387-95332-9 - This book contains an extensive list of references.
- Křížek, Michal; Luca, Florian & Somer, Lawrence (2002), "On the convergence of series of reciprocals of primes related to the Fermat numbers" (PDF), Journal of Number Theory, Elsevier, 97 (1): 95–112, doi:10.1006/jnth.2002.2782, ISSN 0022-314X
- Luca, Florian (2000), "The anti-social Fermat number", American Mathematical Monthly, Mathematical Association of America, 107 (2): 171–173, doi:10.2307/2589441, ISSN 0002-9890
- Ribenboim, Paulo (1996), The New Book of Prime Number Records (3rd ed.), New York: Springer, ISBN 0-387-94457-5
- Robinson, Raphael M. (1954), "Mersenne and Fermat Numbers", Proceedings of the American Mathematical Society, American Mathematical Society, 5 (5): 842–846, doi:10.2307/2031878
- Yabuta, M. (2001), "A simple proof of Carmichael's theorem on primitive divisors" (PDF), Fibonacci Quarterly, Fibonacci Association, 39: 439–443, ISSN 0015-0517
External links
- Fermat prime at Encyclopædia Britannica
- Chris Caldwell, The Prime Glossary: Fermat number at The Prime Pages.
- Luigi Morelli, History of Fermat Numbers
- John Cosgrave, Unification of Mersenne and Fermat Numbers
- Wilfrid Keller, Prime Factors of Fermat Numbers
- Weisstein, Eric W. "Fermat Number". MathWorld.
- Yves Gallot, Generalized Fermat Prime Search
- Mark S. Manasse, Complete factorization of the ninth Fermat number (original announcement)
- Peyton Hayslette, Largest Known Generalized Fermat Prime Announcement