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Fermat number

## Summary

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

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

where n is a non-negative 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, then k must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. As of 2022, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in the OEIS); heuristics suggest that there are no more.

## Basic properties

The Fermat numbers satisfy the following recurrence relations:

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

for n ≥ 1,

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

for n ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > 1. Then a divides both

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

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 that all Fermat numbers are prime. Indeed, the first five Fermat numbers F0, ..., F4 are easily shown to be prime. Fermat's conjecture was refuted by Leonhard Euler in 1732 when he showed that

${\displaystyle F_{5}=2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417.}$

Euler proved that every factor of Fn must have the form k2n+1 + 1 (later improved to k2n+2 + 1 by Lucas).

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 that 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, but little is known about Fermat numbers for 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 of 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 F18233954, and its prime factor 7 × 218233956 + 1, a megaprime, was discovered in October 2020.

### Heuristic arguments

Heuristics suggest that F4 is the last Fermat prime.

The prime number theorem implies that a random integer in a suitable interval around N is prime with probability 1/ln N. If one uses the heuristic that a Fermat number is prime with the same probability as a random integer of its size, and that F5, ..., F32 are composite, then the expected number of Fermat primes beyond F4 (or equivalently, beyond F32) should be

${\displaystyle \sum _{n\geq 33}{\frac {1}{\ln F_{n}}}<{\frac {1}{\ln 2}}\sum _{n\geq 33}{\frac {1}{\log _{2}(2^{2^{n}})}}={\frac {1}{\ln 2}}2^{-32}<3.36\times 10^{-10}.}$

One may interpret this number as an upper bound for the probability that a Fermat prime beyond F4 exists.

This argument is not a rigorous proof. For one thing, it assumes that Fermat numbers behave "randomly", but the factors of Fermat numbers have special properties. Boklan and Conway published a more precise analysis suggesting that the probability that there is another Fermat prime is less than one in a billion.[5]

### Equivalent conditions of primality

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

${\displaystyle F_{n}}$  is prime if and only if ${\displaystyle 3^{(F_{n}-1)/2}\equiv -1{\pmod {F_{n}}}.}$

The expression ${\displaystyle 3^{(F_{n}-1)/2}}$  can be evaluated modulo ${\displaystyle F_{n}}$  by repeated squaring. This makes the test a fast polynomial-time algorithm. But Fermat numbers grow so rapidly that only a handful of them can be tested in a reasonable amount of time and space.

There are some tests for numbers of the form k2m + 1, such as factors of Fermat numbers, for primality.

Proth's theorem (1878). Let ${\displaystyle N}$  = ${\displaystyle k}$ ${\displaystyle 2}$ ${\displaystyle m}$  + ${\displaystyle 1}$  with odd ${\displaystyle k}$  < ${\displaystyle 2}$ ${\displaystyle m}$ . If there is an integer ${\displaystyle a}$  such that
${\displaystyle a^{(N-1)/2}\equiv -1{\pmod {N}}}$
then ${\displaystyle N}$  is prime. Conversely, if the above congruence does not hold, and in addition
${\displaystyle \left({\frac {a}{N}}\right)=-1}$  (See Jacobi symbol)
then ${\displaystyle N}$  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 Fermat numbers' size, it is difficult to factorize or even to check primality. Pépin's test gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The elliptic curve method is a fast method for finding small prime divisors of numbers. Distributed computing project Fermatsearch has found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving Euler's above-mentioned result, proved in 1878 that every factor of the Fermat number ${\displaystyle F_{n}}$ , with n at least 2, is of the form ${\displaystyle k\times 2^{n+2}+1}$  (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 [6]) 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.[4] The distributed computing project Fermat Search is searching for new factors of Fermat numbers.[7] The set of all Fermat factors is A050922 (or, sorted, A023394) in OEIS.

The following factors of Fermat numbers were known before 1950 (since then, digital computers have helped find more factors):

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

As of January 2021, 356 prime factors of Fermat numbers are known, and 312 Fermat numbers are known to be composite.[4] Several new Fermat factors are found each year.[8]

## 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.

${\displaystyle 2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}}$

for all Fermat numbers.

In 1904, Cipolla showed that the product of at least two distinct prime or composite Fermat numbers ${\displaystyle F_{a}F_{b}\dots F_{s},}$  ${\displaystyle a>b>\dots >s>1}$  will be a Fermat pseudoprime to base 2 if and only if ${\displaystyle 2^{s}>a}$ .[9]

## Other theorems about Fermat numbers

Lemma. — If n is a positive integer,

${\displaystyle a^{n}-b^{n}=(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}.}$
Proof

{\displaystyle {\begin{aligned}(a-b)\sum _{k=0}^{n-1}a^{k}b^{n-1-k}&=\sum _{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum _{k=0}^{n-1}a^{k}b^{n-k}\\&=a^{n}+\sum _{k=1}^{n-1}a^{k}b^{n-k}-\sum _{k=1}^{n-1}a^{k}b^{n-k}-b^{n}\\&=a^{n}-b^{n}\end{aligned}}}

Theorem —  If ${\displaystyle 2^{k}+1}$  is an odd prime, then ${\displaystyle k}$  is a power of 2.

Proof

If ${\displaystyle k}$  is a positive integer but not a power of 2, it must have an odd prime factor ${\displaystyle s>2}$ , and we may write ${\displaystyle k=rs}$  where ${\displaystyle 1\leq r .

By the preceding lemma, for positive integer ${\displaystyle m}$ ,

${\displaystyle (a-b)\mid (a^{m}-b^{m})}$

where ${\displaystyle \mid }$  means "evenly divides". Substituting ${\displaystyle a=2^{r},b=-1}$ , and ${\displaystyle m=s}$  and using that ${\displaystyle s}$  is odd,

${\displaystyle (2^{r}+1)\mid (2^{rs}+1),}$

and thus

${\displaystyle (2^{r}+1)\mid (2^{k}+1).}$

Because ${\displaystyle 1<2^{r}+1<2^{k}+1}$ , it follows that ${\displaystyle 2^{k}+1}$  is not prime. Therefore, by contraposition ${\displaystyle k}$  must be a power of 2.

Theorem —  A Fermat prime cannot be a Wieferich prime.

Proof

We show if ${\displaystyle p=2^{m}+1}$  is a Fermat prime (and hence by the above, m is a power of 2), then the congruence ${\displaystyle 2^{p-1}\equiv 1{\bmod {p^{2}}}}$  does not hold.

Since ${\displaystyle 2m|p-1}$  we may write ${\displaystyle p-1=2m\lambda }$ . If the given congruence holds, then ${\displaystyle p^{2}|2^{2m\lambda }-1}$ , and therefore

${\displaystyle 0\equiv {\frac {2^{2m\lambda }-1}{2^{m}+1}}=(2^{m}-1)\left(1+2^{2m}+2^{4m}+\cdots +2^{2(\lambda -1)m}\right)\equiv -2\lambda {\pmod {2^{m}+1}}.}$

Hence ${\displaystyle 2^{m}+1|2\lambda }$ , and therefore ${\displaystyle 2\lambda \geq 2^{m}+1}$ . This leads to ${\displaystyle p-1\geq m(2^{m}+1)}$ , which is impossible since ${\displaystyle m\geq 2}$ .

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

Sketch of proof

Let Gp denote the group of non-zero integers modulo p under multiplication, which has order p−1. Notice that 2 (strictly speaking, its image modulo p) has multiplicative order equal to ${\displaystyle 2^{n+1}}$  in Gp (since ${\displaystyle 2^{2^{n+1}}}$  is the square of ${\displaystyle 2^{2^{n}}}$  which is −1 modulo Fn), so that, by Lagrange's theorem, p − 1 is divisible by ${\displaystyle 2^{n+1}}$  and p has the form ${\displaystyle k2^{n+1}+1}$  for some integer k, as Euler knew. Édouard Lucas went further. Since n > 1, the prime p above is congruent to 1 modulo 8. Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue modulo p, that is, there is integer a such that ${\displaystyle p|a^{2}-2.}$  Then the image of a has order ${\displaystyle 2^{n+2}}$  in the group Gp and (using Lagrange's theorem again), p − 1 is divisible by ${\displaystyle 2^{n+2}}$  and p has the form ${\displaystyle s2^{n+2}+1}$  for some integer s.

In fact, it can be seen directly that 2 is a quadratic residue modulo p, since

${\displaystyle \left(1+2^{2^{n-1}}\right)^{2}\equiv 2^{1+2^{n-1}}{\pmod {p}}.}$

Since an odd power of 2 is a quadratic residue modulo p, so is 2 itself.

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 the Fermat number Fn be P(Fn). Then,

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

## Relationship to constructible polygons

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 that this condition was also necessary, but never published a proof. Pierre Wantzel gave a full proof of necessity in 1837. The result is known as the Gauss–Wantzel theorem:

An n-sided regular polygon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and distinct Fermat primes: in other words, if and only if n is of the form n = 2kp1p2...ps, where k, s are nonnegative integers and the pi are distinct Fermat primes.

A positive integer n is of the above form if and only if its totient φ(n) is a power of 2.

## Applications of Fermat numbers

### Pseudorandom Number Generation

Fermat primes are particularly useful in generating pseudo-random sequences of numbers in the range 1 ... N, where N is a power of 2. The most common method used is to take any seed value between 1 and 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.

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

## Generalized Fermat numbers

Numbers of the form ${\displaystyle a^{2^{\overset {n}{}}}\!\!+b^{2^{\overset {n}{}}}}$  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 = ${\displaystyle 2^{2^{0}}\!+1}$  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 ${\displaystyle a^{2^{\overset {n}{}}}\!\!+1}$  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, ${\displaystyle a^{2^{\overset {n}{}}}\!\!+1}$ , 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 ${\displaystyle F_{n}(a)}$  with ${\displaystyle n>4}$  is ${\displaystyle F_{5}(30)}$ , 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 ${\displaystyle {\frac {a^{2^{n}}\!+1}{2}}}$ , 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 (${\displaystyle F_{n}(a)}$ ) to an even a are ${\displaystyle a^{2^{n}}\!+1}$ , for odd a, they are ${\displaystyle {\frac {a^{2^{n}}\!\!+1}{2}}}$ . 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 ${\displaystyle n}$  such that ${\displaystyle F_{n}(a)}$  is prime, see )

${\displaystyle a}$  numbers ${\displaystyle n}$
such that
${\displaystyle F_{n}(a)}$  is prime
${\displaystyle a}$  numbers ${\displaystyle n}$
such that
${\displaystyle F_{n}(a)}$  is prime
${\displaystyle a}$  numbers ${\displaystyle n}$
such that
${\displaystyle F_{n}(a)}$  is prime
${\displaystyle a}$  numbers ${\displaystyle n}$
such that
${\displaystyle F_{n}(a)}$  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 [13][14] for more information (even bases up to 1000), also see [15] for odd bases)

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

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

${\displaystyle n}$  bases a such that ${\displaystyle F_{n}(a)}$  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 ${\displaystyle F_{n}(a)}$  will be prime for fixed ${\displaystyle n}$ . The number of generalized Fermat primes can be roughly expected to halve as ${\displaystyle n}$  is increased by 1.

### Largest known generalized Fermat primes

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

Rank Prime rank[17] Prime number Generalized Fermat notation Number of digits Discovery date ref.
1 14 10590941048576 + 1 F20(1059094) 6,317,602 Nov 2018 [18]
2 15 9194441048576 + 1 F20(919444) 6,253,210 Sep 2017 [19]
3 51 3638450524288 + 1 F19(3638450) 3,439,810 May 2020 [20]
4 53 9 ×211366286 + 1 F1(3 × 25683143) 3,421,594 Mar 2020 [21]
5 55 3214654524288 + 1 F19(3214654) 3,411,613 Dec 2019 [22]

On the Prime Pages one can find the current top 100 generalized Fermat primes.

## Notes

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 Keller, Wilfrid (January 18, 2021), "Prime Factors of Fermat Numbers", ProthSearch.com, retrieved January 19, 2021
5. ^ Boklan, Kent D.; Conway, John H. (2017). "Expect at most one billionth of a new Fermat Prime!". 39 (1): 3–5. arXiv:1605.01371. {{cite journal}}: Cite journal requires |journal= (help)
6. ^ Sandifer, Ed. "How Euler Did it" (PDF). MAA Online. Mathematical Association of America. Retrieved 2020-06-13.
7. ^ ":: F E R M A T S E A R C H . O R G :: Home page". www.fermatsearch.org. Retrieved 7 April 2018.
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. ISBN 9780387218502. 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 11 July 2019.
17. ^ Caldwell, Chris K. "Database Search Output". The Prime Pages. Retrieved 11 July 2019.
18. ^ 10590941048576 + 1
19. ^ 9194441048576 + 1
20. ^ 3638450524288 + 1
21. ^ 9 · 211366286 + 1
22. ^ 3214654524288 + 1

## References

• Golomb, S. W. (January 1, 1963), "On the sum of the reciprocals of the Fermat numbers and related irrationalities", Canadian Journal of Mathematics, 15: 475–478, doi:10.4153/CJM-1963-051-0, S2CID 123138118
• Grytczuk, A.; Luca, F. & Wójtowicz, M. (2001), "Another note on the greatest prime factors of Fermat numbers", Southeast Asian Bulletin of Mathematics, 25 (1): 111–115, doi:10.1007/s10012-001-0111-4, S2CID 122332537
• Guy, Richard K. (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics, vol. 1 (3rd ed.), New York: Springer Verlag, pp. A3, A12, B21, ISBN 978-0-387-20860-2
• Křížek, Michal; Luca, Florian & Somer, Lawrence (2001), 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS books in mathematics, vol. 10, New York: Springer, ISBN 978-0-387-95332-8 - 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", Journal of Number Theory, 97 (1): 95–112, doi:10.1006/jnth.2002.2782
• Luca, Florian (2000), "The anti-social Fermat number", American Mathematical Monthly, 107 (2): 171–173, doi:10.2307/2589441, JSTOR 2589441
• Ribenboim, Paulo (1996), The New Book of Prime Number Records (3rd ed.), New York: Springer, ISBN 978-0-387-94457-9
• Robinson, Raphael M. (1954), "Mersenne and Fermat Numbers", Proceedings of the American Mathematical Society, 5 (5): 842–846, doi:10.2307/2031878, JSTOR 2031878
• Yabuta, M. (2001), "A simple proof of Carmichael's theorem on primitive divisors" (PDF), Fibonacci Quarterly, 39: 439–443