# Fermat number

In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form

$F_{n}=2^{2^{n}}+1,$ Named after Pierre de Fermat 5 5 Fermat numbers 3, 5, 17, 257, 65537 65537 A019434

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 2019, the only known Fermat primes are F0, F1, F2, F3, and F4 (sequence A019434 in the OEIS).

## Basic properties

The Fermat numbers satisfy the following recurrence relations:

$F_{n}=(F_{n-1}-1)^{2}+1$ $F_{n}=F_{0}\cdots F_{n-1}+2$ for n ≥ 1,

$F_{n}=F_{n-1}+2^{2^{n-1}}F_{0}\cdots F_{n-2}$ $F_{n}=F_{n-1}^{2}-2(F_{n-2}-1)^{2}$ 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

$F_{0}\cdots F_{j-1}$ 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.

## 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 F0,...,F4 are easily shown to be prime. However, the conjecture was refuted by Leonhard Euler in 1732 when he showed 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 k2n+1 + 1 (later improved to k2n+2 + 1 by Lucas).

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 (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. 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. 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. 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.

### 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(n). Therefore, the total expected number of Fermat primes is at most

{\begin{aligned}\sum _{n=0}^{\infty }{\frac {1}{\ln F_{n}}}&={\frac {1}{\ln 2}}\sum _{n=0}^{\infty }{\frac {1}{\log _{2}\left(2^{2^{n}}+1\right)}}\\&<{\frac {1}{\ln 2}}\sum _{n=0}^{\infty }2^{-n}\\&={\frac {2}{\ln 2}}\\&=2.885...\end{aligned}} 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

{\begin{aligned}A\sum _{n=0}^{\infty }{\frac {\ln 2^{n+1}}{\ln F_{n}}}&=A\sum _{n=0}^{\infty }{\frac {\log _{2}2^{n+1}}{\log _{2}\left(2^{2^{n}}+1\right)}}\\& ### Equivalent conditions of primality

Let $F_{n}=2^{2^{n}}+1$ be the nth Fermat number. Pépin's test states that for n > 0,

$F_{n}$ 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. 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
$a^{(N-1)/2}\equiv -1{\pmod {N}}$ 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 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

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 $F_{n}$ , with n at least 2, is of the form $k\times 2^{n+2}+1$ (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 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 = 13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,097 (155 digits) = 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. 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.

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 (since the '50s, digital computers have helped find more factors):

Year Finder Fermat number Factor
1732 Euler $F_{5}$ $5\cdot 2^{7}+1$ 1732 Euler $F_{5}$ (fully factored) $52347\cdot 2^{7}+1$ 1855 Clausen $F_{6}$ $1071\cdot 2^{8}+1$ 1855 Clausen $F_{6}$ (fully factored) $262814145745\cdot 2^{8}+1$ 1877 Pervushin $F_{12}$ $7\cdot 2^{14}+1$ 1878 Pervushin $F_{23}$ $5\cdot 2^{25}+1$ 1886 Seelhoff $F_{36}$ $5\cdot 2^{39}+1$ 1899 Cunningham $F_{11}$ $39\cdot 2^{13}+1$ 1899 Cunningham $F_{11}$ $119\cdot 2^{13}+1$ 1903 Western $F_{9}$ $37\cdot 2^{16}+1$ 1903 Western $F_{12}$ $397\cdot 2^{16}+1$ 1903 Western $F_{12}$ $973\cdot 2^{16}+1$ 1903 Western $F_{18}$ $13\cdot 2^{20}+1$ 1903 Cullen $F_{38}$ $3\cdot 2^{41}+1$ 1906 Morehead $F_{73}$ $5\cdot 2^{75}+1$ 1925 Kraitchik $F_{15}$ $579\cdot 2^{21}+1$ As of March 2019, 349 prime factors of Fermat numbers are known, and 305 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

Lemma.  If n is a positive integer,

$a^{n}-b^{n}=(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}.$ Theorem   If $2^{k}+1$ is an odd prime, then $k$ is a power of 2.

Theorem   A Fermat prime cannot be a Wieferich prime.

Theorem    Any prime divisor p of $F_{n}=2^{2^{n}}+1$ is of the form $k2^{n+2}+1$ whenever n > 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 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

### 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.

$V_{j+1}=\left(A\times V_{j}\right){\bmod {P}}$ (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

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.

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

$P(F_{n})\geq 2^{n+2}(4n+9)+1.$ (Grytczuk, Luca & Wójtowicz 2001)

## Generalized Fermat numbers

Numbers of the form $a^{2^{\overset {n}{}}}+b^{2^{\overset {n}{}}}$ 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 = $2^{2^{0}}+1$ is not a counterexample.)

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

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.

If we require n > 0, then Landau's fourth problem asks if there are infinitely many generalized Fermat primes Fn(a).

### 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 a, because if a 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 the list, the generalized Fermat numbers ($F_{n}(a)$ ) to an even a are $a^{2^{n}}+1$ , for odd a, they are ${\frac {a^{2^{n}}+1}{2}}$ . 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 $n$ such that $F_{n}(a)$ is prime, see )

$a$ numbers $n$ such that
$F_{n}(a)$ is prime
$a$ numbers $n$ such that
$F_{n}(a)$ is prime
$a$ numbers $n$ such that
$F_{n}(a)$ is prime
$a$ numbers $n$ such that
$F_{n}(a)$ 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  for more information (even bases up to 1000), also see  for odd bases)

(For the smallest prime of the form $F_{n}(a,b)$ (for odd $a+b$ ), see also )

$a$ $b$ numbers $n$ such that
${\frac {a^{2^{n}}+b^{2^{n}}}{\gcd(a+b,2)}}(=F_{n}(a,b))$ 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 $F_{n}(a)$ is prime, see )

$n$ bases a such that $F_{n}(a)$ 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, 1059094, ... A321323

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 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.

### Largest known generalized Fermat primes

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

Rank Prime rank Prime number Generalized Fermat notation Number of digits Found date ref.
1 14 10590941048576 + 1 F20(1059094) 6,317,602 Nov 2018 
2 15 9194441048576 + 1 F20(919444) 6,253,210 Sep 2017 
3 26 2877652524288 + 1 F19(2877652) 3,386,397 Jun 2019 
4 27 2788032524288 + 1 F19(2788032) 3,379,193 Apr 2019 
5 28 2733014524288 + 1 F19(2733014) 3,374,655 Mar 2019 

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