Fermat number

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Fermat prime
Named afterPierre de Fermat
No. of known terms5
Conjectured no. of terms5
Subsequence ofFermat numbers
First terms3, 5, 17, 257, 65537
Largest known term65537
OEIS indexA019434

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:

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … (sequence A000215 in the OEIS).

If 2k + 1 is prime, and k > 0, it can be shown that k must be a power of two. (If k = ab where 1 ≤ a, bk and b is odd, then 2k + 1 = (2a)b + 1 ≡ (−1)b + 1 = 0 (mod 2a + 1). See below for a complete proof.) In other words, every prime of the form 2k + 1 (other than 2 = 20 + 1) is a Fermat number, and such primes are called Fermat primes. As of 2018, the only known Fermat primes are F0, F1, F2, F3, and F4 (sequence A019434 in the OEIS).

Basic properties[edit]

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 Fi and Fj have a common factor a > 1. Then a divides both

and Fj; 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 Fn, choose a prime factor pn; then the sequence {pn} is an infinite sequence of distinct primes.

Further properties:

Primality of Fermat numbers[edit]

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 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 Fn must have the form k2n+1 + 1 (later improved to k2n+2 + 1 by Lucas).

The fact that 641 is a factor of F5 can be easily deduced 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 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 Fn with n > 4. However, little is known about Fermat numbers with large n.[2] In fact, each of the following is an open problem:

As of 2014, it is known that Fn is composite for 5 ≤ n ≤ 32, although amongst these, complete factorizations of Fn 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 F3329780, and its prime factor 193 × 23329782 + 1, a megaprime, was discovered by the PrimeGrid collaboration in July 2014.[4][5]

Heuristic arguments for density[edit]

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.

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 Fn exceeds 2n + 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 = 2n + 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[edit]

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 are some tests that can be used to test numbers of the form k2m + 1, such as factors of Fermat numbers, for primality.

Proth's theorem (1878). Let N = k2m + 1 with odd k < 2m. 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 = Fn > 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[edit]

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:

F0 = 21 + 1 = 3 is prime
F1 = 22 + 1 = 5 is prime
F2 = 24 + 1 = 17 is prime
F3 = 28 + 1 = 257 is prime
F4 = 216 + 1 = 65,537 is the largest known Fermat prime
F5 = 232 + 1 = 4,294,967,297
= 641 × 6,700,417 (fully factored 1732)
F6 = 264 + 1 = 18,446,744,073,709,551,617 (20 digits)
= 274,177 × 67,280,421,310,721 (14 digits) (fully factored 1855)
F7 = 2128 + 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) (fully factored 1970)
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)
= 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)
F9 = 2512 + 1 = 13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,0
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) (fully factored 1990)
F10 = 21024 + 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) (fully factored 1995)
F11 = 22048 + 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) (fully factored 1988)

As of 2018, only F0 to F11 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.

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.[7]

The following factors of Fermat numbers were known before 1950 (since the '50s, 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 April 2018, 342 prime factors of Fermat numbers are known, and 298 Fermat numbers are known to be composite.[4] Several new Fermat factors are found each year.[8]

Pseudoprimes and Fermat numbers[edit]

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 .[9]

Other theorems about Fermat numbers[edit]

Lemma. — If n is a positive integer,

Theorem —  If is an odd prime, then is a power of 2.

Theorem —  A Fermat prime cannot be a Wieferich prime.

Theorem (Édouard Lucas) —  Any prime divisor p of is of the form whenever n > 1.

Relationship to constructible polygons[edit]

Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)

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 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:

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 = 2kp1p2ps, where k is a nonnegative integer 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[edit]

Pseudorandom Number Generation[edit]

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.

(see Linear congruential generator, RANDU)

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 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[edit]

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 nn + 1 is prime, there exists an integer m such that n = 22m. The equation nn + 1 = F(2m+m) holds in that case.[10][11]

Let the largest prime factor of Fermat number Fn be P(Fn). Then,

(Grytczuk, Luca & Wójtowicz 2001)

Generalized Fermat numbers[edit]

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.)

An example of a probable prime of this form is 12465536 + 5765536 (found by Valeryi Kuryshev).[12]

By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form 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, , 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 Fn(a).

Generalized Fermat primes[edit]

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 a, because if a 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 (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 a are , for odd a, they are . If a is a perfect power with an odd exponent (sequence A070265 in the OEIS), then all generalized Fermat number can be algebraic factored, so they cannot be prime)

(For the smallest number such that is prime, see OEISA253242)

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, ...
b known generalized (half) 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 (not possible)
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 (not possible)
28 29, 614657
29 421, 353641, 125123236840173674393761
30 31, 185302018885184100000000000000000000000000000001
31
32 (not possible)
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 (852 digits: 214787904487...289480994817)
47 11905643330881
48 5308417
49 1201
50

(See [13][14] for more information (even bases up to 1000), also see [15] for odd bases)

(For the smallest prime of the form (for odd ), see also OEISA111635)

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 a such that is prime, see OEISA056993)

bases a such that is prime (only consider even a) 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, 1828858, 2042774, 2514168, 2611294, 2676404, 3060772, ... A244150
19 75898, 341112, 356926, 475856, 1880370, 2061748, 2312092,... A243959
20 919444, ...

The smallest base b such that b2n + 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, ... (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 OEISA228101 and OEISA084712)

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[edit]

The following is a list of the 5 largest known generalized Fermat primes.[16] They are all megaprimes. As of September 2018 the whole top-5 was discovered by participants in the PrimeGrid project.

Rank Prime rank[17] Prime number Generalized Fermat notation Number of digits Found date ref.
1 13 9194441048576 + 1 F20(919444) 6,253,210 Sep 2017 [18]
2 24 2312092524288 + 1 F19(2312092) 3,336,572 Aug 2018 [19]
3 25 2061748524288 + 1 F19(2061748) 3,310,478 Mar 2018 [20]
4 26 1880370524288 + 1 F19(1880370) 3,289,511 Jan 2018 [21]
5 30 475856524288 + 1 F19(475856) 2,976,633 Aug 2012 [22]

On the Prime Pages you can perform a search yielding the current top 100 generalized Fermat primes.

See also[edit]

Notes[edit]

  1. ^ Křížek, Luca & Somer 2001, p. 38, Remark 4.15
  2. ^ Chris Caldwell, "Prime Links++: special forms" Archived 2013-12-24 at the Wayback Machine. at The Prime Pages.
  3. ^ Ribenboim 1996, p. 88.
  4. ^ a b c d Keller, Wilfrid (February 7, 2012), "Prime Factors of Fermat Numbers", ProthSearch.com, retrieved January 14, 2017
  5. ^ "PrimeGrid's Mega Prime Search – 193*2^3329782+1 (official announcement)" (PDF). PrimeGrid. Retrieved 7 August 2014.
  6. ^ ":: F E R M A T S E A R C H . O R G :: Home page". www.fermatsearch.org. Retrieved 7 April 2018.
  7. ^ Boklan, Kent D.; Conway, John H. (2016). "Expect at most one billionth of a new Fermat Prime!". arXiv:1605.01371 [math.NT].
  8. ^ ":: F E R M A T S E A R C H . O R G :: News". www.fermatsearch.org. Retrieved 7 April 2018.
  9. ^ Krizek, Michal; Luca, Florian; Somer, Lawrence (14 March 2013). "17 Lectures on Fermat Numbers: From Number Theory to Geometry". Springer Science & Business Media. Retrieved 7 April 2018 – via Google Books.
  10. ^ Jeppe Stig Nielsen, "S(n) = n^n + 1".
  11. ^ Weisstein, Eric W. "Sierpiński Number of the First Kind". MathWorld.
  12. ^ PRP Top Records, search for x^(2^16)+y^(2^16), by Henri & Renaud Lifchitz.
  13. ^ "Generalized Fermat Primes". jeppesn.dk. Retrieved 7 April 2018.
  14. ^ "Generalized Fermat primes for bases up to 1030". noprimeleftbehind.net. Retrieved 7 April 2018.
  15. ^ "Generalized Fermat primes in odd bases". fermatquotient.com. Retrieved 7 April 2018.
  16. ^ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
  17. ^ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
  18. ^ 9194441048576 + 1
  19. ^ 2312092524288 + 1
  20. ^ 2061748524288 + 1
  21. ^ 1880370524288 + 1
  22. ^ 475856524288 + 1

References[edit]

External links[edit]