Fermat number

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 non-negative 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. ^b + 1 ≡ ^b + 1 = 0 In other words, every prime of the form 2^k + 1 is a Fermat number, and such primes are called Fermat primes. As of 2019, the only known Fermat primes are F₀, F₁, F₂, F₃, and F₄.

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 second equation, we can deduce Goldbach's theorem : 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 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 digit of a Fermat number is 7.
The sum of the reciprocals of all the Fermat numbers is irrational.
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, the conjecture was refuted by Leonhard Euler in 1732 when he showed that
Euler proved that every factor of F_n must have the form k2ⁿ⁺¹ + 1.
The fact that 641 is a factor of F₅ can be deduced from the equalities 641 = 2⁷ × 5 + 1 and 641 = 2⁴ + 5⁴. It follows from the first equality that 2⁷ × 5 ≡ −1 and therefore that 2²⁸ × 5⁴ ≡ 1 . On the other hand, the second equality implies that 5⁴ ≡ −2⁴ . These congruences imply that 2³² ≡ −1 .
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. However, 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?
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 amongst 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_5523858, and its prime factor, a megaprime, was discovered by the PrimeGrid collaboration by S. Brown, Reynolds, Penné & Fougeron in January 2020.

Heuristic arguments for density

There are several probabilistic arguments for the finitude of Fermat primes.
According to the prime number theorem, the "probability" that a number n is prime is about 1/ln. 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.
If we regard the conditional probability that n is prime, given that we know all its prime factors exceed B, as at most A ln / ln, then using Euler's theorem that the least prime factor of F_n exceeds , we would find instead

Equivalent conditions of primality

Let be the nth Fermat number. Pépin's test states that for n > 0,
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 are some tests that can be used to test numbers of the form k2^m + 1, such as factors of Fermat numbers, for primality.
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 the Fermat number, with n at least 2, is of the form , 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 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 in OEIS.
It is possible that the only primes of this form are 3, 5, 17, 257 and 65,537. Indeed, Boklan and Conway published in 2016 a very precise analysis suggesting that the probability of the existence of another Fermat prime is less than one in a billion.
The following factors of Fermat numbers were known before 1950 :

Year	Finder	Fermat number	Factor
1732	Euler
1732	Euler
1855	Clausen
1855	Clausen
1877	Pervushin
1878	Pervushin
1886	Seelhoff
1899	Cunningham
1899	Cunningham
1903	Western
1903	Western
1903	Western
1903	Western
1903	Cullen
1906	Morehead
1925	Kraitchik

, 351 prime factors of Fermat numbers are known, and 307 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.
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

Relationship to constructible polygons

developed the theory of Gaussian periods in his Disquisitiones Arithmeticae and formulated a sufficient condition for the constructibility of regular polygons. Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem:
A positive integer n is of the above form if and only if its totient φ 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. 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 possible values. 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.
The series of reciprocals of all prime divisors of Fermat numbers is convergent.
If nⁿ + 1 is prime, there exists an integer m such that n = 2²^{^m}. The equation
nⁿ + 1 = F
holds in that case.
Let the largest prime factor of the Fermat number F_n be P. 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.
An example of a probable prime of this form is 124⁶⁵⁵³⁶ + 57⁶⁵⁵³⁶.
By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form as F_n. In this notation, for instance, the number 100,000,001 would be written as F₃. 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.

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 with is, or 30³² + 1. Besides, we can define "half generalized Fermat numbers" for an odd base, a half generalized Fermat number to base a is, and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base.

	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		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		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		48	2,...	64
17	2,...	33	0, 3,...	49	1,...	65	1, 2, 5,...

b	known generalized Fermat prime base b
2	3, 5, 17, 257, 65537
3	2, 5, 41, 21523361, 926510094425921, 1716841910146256242328924544641
4	5, 17, 257, 65537
5	3, 13, 313
6	7, 37, 1297
7	1201
8
9	5, 41, 21523361, 926510094425921, 1716841910146256242328924544641
10	11, 101
11	61, 7321
12	13
13	7, 14281, 407865361
14	197
15	113
16	17, 257, 65537
17	41761
18	19
19	181
20	401, 160001
21	11, 97241, 1023263388750334684164671319051311082339521
22	23
23	139921
24	577, 331777
25	13, 313
26	677
27
28	29, 614657
29	421, 353641, 125123236840173674393761
30	31, 185302018885184100000000000000000000000000000001
31
32
33	17, 703204309121
34	1336337
35	613, 750313, 330616742651687834074918381127337110499579842147487712949050636668246738736343104392290115356445313
36	37, 1297
37	19
38
39	761, 1156721
40	41, 1601
41	31879515457326527173216321
42	43
43	5844100138801
44	197352587024076973231046657
45	23, 1013
46	47, 4477457, 46⁵¹²+1
47	11905643330881
48	5308417
49	1201
50

		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
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,...

	bases such that is prime	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,...
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,...
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,...
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,...
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,...
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,...
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,...
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,...
8	278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332,...
9	46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992,...
10	824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670,...
11	150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654,...
12	1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696,...
13	30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894, 176206, 180924, 201170, 212724, 222764, 225174, 241600,...
14	67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, 422666, 426690, 449732, 462470, 468144, 498904, 506664,...
15	70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, 553602, 743788, 825324, 831648, 855124, 999236, 1041870,...
16	48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540,...
17	62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192,...
18	24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, 1828858, 2042774, 2514168, 2611294, 2676404, 3060772,...
19	75898, 341112, 356926, 475856, 1880370, 2061748, 2312092,...
20	919444, 1059094,...

The smallest base b such that b^2ⁿ + 1 is prime are
The smallest k such that ^k + 1 is prime are
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 5 largest known generalized Fermat primes. They are all megaprimes. The whole top-5 is discovered by participants in the PrimeGrid project.

Rank	Prime rank	Prime number	Generalized Fermat notation	Number of digits	Found date	ref.
1	14	1059094^1048576 + 1	F₂₀	6,317,602	Nov 2018
2	15	919444^1048576 + 1	F₂₀	6,253,210	Sep 2017
3	31	3214654⁵²⁴²⁸⁸ + 1	F₁₉	3,411,613	Dec 2019
4	32	2985036⁵²⁴²⁸⁸ + 1	F₁₉	3,394,739	Sep 2019
5	33	2877652⁵²⁴²⁸⁸ + 1	F₁₉	3,386,397	Jun 2019

On the Prime Pages you can perform a search yielding the .

Popular movies

The Hunger Games (film) - 2012 American dystopian action thriller science fiction-adventure film directed by Gary Ross and based on Suzanne Collins’s 2008 novel of the same name. It is the first insta...
untitled Captain Marvel sequel - part of Marvel Cinematic Universe....
Killers of the Flower Moon (film project) - Killers of the Flower Moon - film project in United States of America. It was presented as drama, detective fiction, thriller. The film project starred Leonardo Dicaprio, Robert De Niro. Director of...
Five Nights at Freddy's (film) - Five Nights at Freddy's - film published in 2017 in United States of America. Scenarist of the film - Scott Cawthon....

Popular books

Book of Revelation - The Book of Revelation is the final book of the New Testament, and consequently is also the final book of the Christian Bible. Its title is derived from the first word of the Koine Greek text: apok...
Book of Genesis - account of the creation of the world, the early history of humanity, Israel's ancestors and the origins...
Gospel of Matthew - The Gospel According to Matthew is the first book of the New Testament and one of the three synoptic gospels. It tells how Israel's Messiah, rejected and executed in Israel, pronounces judgement on ...
Michelin Guide - Michelin Guides are a series of guide books published by the French tyre company Michelin for more than a century. The term normally refers to the annually published Michelin Red Guide , the oldest...
Psalms - The Book of Psalms , commonly referred to simply as Psalms , the Psalter or "the Psalms", is the first book of the Ketuvim , the third section of the Hebrew Bible, and thus a book of th...
Ecclesiastes - Ecclesiastes is one of 24 books of the Tanakh , where it is classified as one of the Ketuvim . Originally written c. 450–200 BCE, it is also among the canonical Wisdom literature of the Old Tes...
The 48 Laws of Power - non-fiction book by American author Robert Greene. The book...

Popular television series

The Crown (TV series) - historical drama web television series about the reign of Queen Elizabeth II, created and principally written by Peter Morgan, and produced by Left Bank Pictures and Sony Pictures Tel...
Friends - American sitcom television series, created by David Crane and Marta Kauffman, which aired on NBC from September 22, 1994, to May 6, 2004, lasting ten seasons. With an ensemble cast sta...
Young Sheldon - spin-off prequel to The Big Bang Theory and begins with the character Sheldon...
Modern Family - American television mockumentary family sitcom created by Christopher Lloyd and Steven Levitan for the American Broadcasting Company. It ran for eleven seasons, from September 23...
Loki (TV series) - upcoming American web television miniseries created for Disney+ by Michael Waldron, based on the Marvel Comics character of the same name. It is set in the Marvel Cinematic Universe, shar...
Game of Thrones - American fantasy drama television series created by David Benioff and D. B. Weiss for HBO. It...
Shameless (American TV series) - American comedy-drama television series developed by John Wells which debuted on Showtime on January 9, 2011. It...