Fermat number

Fermat prime
Named after	Pierre de Fermat
Number of known terms	5
Conjectured number of terms	5
Subsequence of	Fermat numbers
First terms	3, 5, 17, 257, 65537
Largest known term	65537
OEIS index	A019434

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

F_{n}=2^{2^{n}}+1,

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 2^k + 1 is prime, and k > 0, it can be shown that k must be a power of two. (If k = ab where 1 ≤ a, b ≤ k and b is odd, then 2^k + 1 = (2^a)^b + 1 ≡ (−1)^b + 1 = 0 (mod 2^a + 1). See below for a complete proof.) In other words, every prime of the form 2^k + 1 (other than 2 = 2⁰ + 1) is a Fermat number, and such primes are called Fermat primes. As of 2016, 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\!

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\!

F_{n} = F_{0} \cdots F_{n-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 F_i and F_j have a common factor a > 1. Then a divides both

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

and F_j; hence a divides their difference, 2. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each F_n, choose a prime factor p_n; then the sequence {p_n} is an infinite sequence of distinct primes.

Further properties:

No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime.
With the exception of F₀ and F₁, the last digit of a Fermat number is 7.
The sum of the reciprocals of all the Fermat numbers (sequence A051158 in the OEIS) is irrational. (Solomon W. Golomb, 1963)

Primality of Fermat numbers

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured (but admitted he could not prove) that all Fermat numbers are prime. Indeed, the first five Fermat numbers F₀,...,F₄ are easily shown to be prime. However, this 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 F_n 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 F_n with n > 4. However, little is known about Fermat numbers with large n.^[2] In fact, each of the following is an open problem:

Is F_n composite for all n > 4?
Are there infinitely many Fermat primes? (Eisenstein 1844)^[3]
Are there infinitely many composite Fermat numbers?
Does a Fermat number exist that is not square-free?

As of 2014, it is known that F_n is composite for 5 ≤ n ≤ 32, although amongst these, complete factorizations of F_n are known only for 0 ≤ n ≤ 11, and there are no known prime factors for n = 20 and n = 24.^[4] The largest Fermat number known to be composite is F_3329780, and its prime factor 193 × 2^3329782 + 1, a megaprime, was discovered by the PrimeGrid collaboration in July 2014.^[4]^[5]

Heuristic arguments for density

There are several probabilistic arguments for estimating the number of Fermat primes. However these arguments give quite different estimates, depending on how much information about Fermat numbers one uses, and some predict no further Fermat primes while others predict infinitely many Fermat primes.

The following heuristic argument suggests there are only finitely many Fermat primes: according to the prime number theorem, the "probability" that a number n is prime is about 1/ln(n). Therefore, the total expected number of Fermat primes is at most

{\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}}.\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. For example, a similar argument applied to the numbers of the form 2ⁿ+1 suggests that infinitely many of them should be prime, which would mean that there are an infinite number of Fermat primes (though much the same argument applied to the numbers 2ⁿ suggests the ridiculous result that infinitely many of them should be prime, so such probability arguments should not be taken too seriously).

If (more sophisticatedly) we regard the conditional probability that n is prime, given that we know all its prime factors exceed B, as at most A ln(B) / ln(n), then using Euler's theorem that the least prime factor of F_n exceeds 2^{n + 1}, we would find instead

{\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)}}\\&<A\sum _{n=0}^{\infty }(n+1)2^{-n}\\&=4A.\end{aligned}}

Although such arguments engender the belief that there are only finitely many Fermat primes, one can also produce arguments for the opposite conclusion. Suppose we regard the conditional probability that n is prime, given that we know all its prime factors are 1 modulo M, as at least CM/ln(n). Then using Euler's result that M = 2^n + 1 we would find that the expected total number of Fermat primes was at least

{\begin{aligned}C\sum _{n=0}^{\infty }{\frac {2^{n+1}}{\ln F_{n}}}&={\frac {C}{\ln 2}}\sum _{n=0}^{\infty }{\frac {2^{n+1}}{\log _{2}\left(2^{2^{n}}+1\right)}}\\&>{\frac {C}{\ln 2}}\sum _{n=0}^{\infty }1\\&=\infty ,\end{aligned}}

and indeed this argument predicts that an asymptotically constant fraction of Fermat numbers are prime.

Equivalent conditions of primality

Main article: Pépin's test

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 some tests that can be used to test numbers of the form k2^m + 1, such as factors of Fermat numbers, for primality.

Proth's theorem (1878)—Let N = k2^m + 1 with odd k < 2^m. If there is an integer a such that

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 = F_n > 3, then the above Jacobi symbol is always equal to −1 for a = 3, and this special case of Proth's theorem is known as Pépin's test. Although Pépin's test and Proth's theorem have been implemented on computers to prove the compositeness of some Fermat numbers, neither test gives a specific nontrivial factor. In fact, no specific prime factors are known for n = 20 and 24.

Factorization of Fermat numbers

Because of the size of Fermat numbers, it is difficult to factorize or even to check primality. Pépin's test gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The elliptic curve method is a fast method for finding small prime divisors of numbers. Distributed computing project Fermatsearch has successfully found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving the above-mentioned result by Euler, proved in 1878 that every factor of Fermat number $F_{n}$ , with n at least 2, is of the form $k\times2^{n+2}+1$ (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
F₆	=	2⁶⁴	+	1	=	18,446,744,073,709,551,617 (20 digits)
					=	274,177 × 67,280,421,310,721 (14 digits)
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)
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)
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)
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)
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)

As of 2016, 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.

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

Year	Finder	Fermat number	Factor
1732	Euler	$F_5$	$5 \cdot 2^7 + 1$
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 September 2016, 331 prime factors of Fermat numbers are known, and 288 Fermat numbers are known to be composite.^[4]

Pseudoprimes and Fermat numbers

Like composite numbers of the form 2^p − 1, every composite Fermat number is a strong pseudoprime to base 2. This is because all strong pseudoprimes to base 2 are also Fermat pseudoprimes - i.e.

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 $2^{s}>a$ .^[7]

Selfridge's Conjecture

John L. Selfridge made an conjecture. Let g(n) be the number of distinct prime factors of 2^2ⁿ + 1 (sequence A046052 in the OEIS). Then g(n) is not monotonic (nondecreasing). If another Fermat prime exists, that would imply the conjecture.^[8]

Other theorems about Fermat numbers

Lemma: If n is a positive integer,

a^n-b^n=(a-b)\sum_{k=0}^{n-1} a^kb^{n-1-k}.

Proof:

(a-b)\sum_{k=0}^{n-1}a^kb^{n-1-k}

=\sum_{k=0}^{n-1}a^{k+1}b^{n-1-k}-\sum_{k=0}^{n-1}a^kb^{n-k}

=a^n+\sum_{k=1}^{n-1}a^kb^{n-k}-\sum_{k=1}^{n-1}a^kb^{n-k}-b^n

=a^n-b^n.

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

Proof:

If $k$ is a positive integer but not a power of 2, $k$ must have an odd prime factor, $s > 2$ .

This means that $k=rs$ where $1\leq r<k$ .

By the preceding lemma, for positive integer $m$ ,

(a-b) \mid (a^m-b^m)

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

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

and thus

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

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

Theorem: A Fermat prime cannot be a Wieferich prime.

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

It is easy to show $2m |p-1$ . Now write $p-1=2m\lambda$ . If the given congruence holds, then $p^2|2^{2m\lambda}-1$ , and therefore

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

Hence $2^m+1|2\lambda$ ,and therefore $2\lambda \geq 2^m+1$ . This leads to

$p-1 \geq m(2^m+1)$ , which is impossible since $m \geq 2$ .

A theorem of Édouard Lucas: Any prime divisor p of F_n = $2^{2^{\overset{n}{}}}+1$ is of the form $k2^{n+2}+1$ whenever n is greater than one.

Sketch of proof:

Let G_p denote the group of non-zero elements of the integers (mod p) under multiplication, which has order p-1. Notice that 2 (strictly speaking, its image (mod p)) has multiplicative order dividing $2^{n+1}$ in G_p (since $2^{2^{\overset{n+1}{}}}$ is the square of $2^{2^{\overset{n}{}}}$ which is -1 mod F_n), so that, by Lagrange's theorem, p-1 is divisible by $2^{n+1}$ and p has the form $k2^{n+1}+1$ for some integer k, as Euler knew. Édouard Lucas went further. Since n is greater than 1, the prime p above is congruent to 1 (mod 8). Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue (mod p), that is, there is integer a such that a² -2 is divisible by p. Then the image of a has order $2^{n+2}$ in the group G_p and (using Lagrange's theorem again), p-1 is divisible by $2^{n+2}$ and p has the form $s2^{n+2}+1$ for some integer s.

In fact, it can be seen directly that 2 is a quadratic residue (mod p), since $\left(1+2^{2^{n-1}}\right)^{2}\equiv 2^{1+2^{n-1}}$ (mod p). Since an odd power of 2 is a quadratic residue (mod p), so is 2 itself.

Relationship to constructible polygons

Main article: Constructible polygon

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

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

Applications of Fermat numbers

Pseudorandom Number Generation

Fermat primes are particularly useful in generating pseudo-random sequences of numbers in the range 1 … N, where N is a power of 2. The most common method used is to take any seed value between 1 and P − 1, where P is a Fermat prime. Now multiply this by a number A, which is greater than the square root of P and is a primitive root modulo P (i.e., it is not a quadratic residue). Then take the result modulo P. The result is the new value for the RNG.

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 2^X 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 = 2²^{^m}. The equation nⁿ + 1 = F₍₂_^m_+m) holds at that time.^[9]

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

P(F_n)\ge 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.)

By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form $a^{2^{ \overset{n} {}}} + 1$ as F_n(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 F_n(a).

Generalized Fermat primes

Because of the ease of proving their primality, generalized Fermat primes have become in recent years a hot topic for research within the field of number theory. Many of the largest known primes today are generalized Fermat primes.

Generalized Fermat numbers can be prime only for even $a$ , 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 algrabic factored, so they cannot be prime)

(For the smallest number $n$ such that $F_n(a)$ is prime, see A253242)

$a$	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, ...

(See ^[10]^[11] for more information (even bases up to 1000), also see ^[12] for odd bases)

(For the smallest prime of the form $F_n(a,b)$ (for odd $a+b$ ), see also A111635)

$a$	$b$	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 A056993)

$n$	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, ...	A244150
19	75898, 341112, 356926, 475856, ...	A243959

The smallest base b such that b^2ⁿ + 1 is prime are

2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, ... (sequence A056993 in the OEIS)

The smallest k such that (2n)^k + 1 is prime are

1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 4, 1, ... (The next term is unknown) (sequence A079706 in the OEIS) (also see

A228101 and

A084712)

A more elaborate theory can be used to predict the number of bases for which $F_n(a)$ 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 10 largest known generalized Fermat primes.^[13] They are all megaprimes. As of April 2015 the whole top-10 was discovered by participants in the PrimeGrid project.

Rank	Prime rank^[14]	Prime number	Generalized Fermat notation	Number of digits	Found date	reference
1	16	475856⁵²⁴²⁸⁸ + 1	F₁₉(475856)	2,976,633	2012 August 8	^[15]
2	17	356926⁵²⁴²⁸⁸ + 1	F₁₉(356926)	2,911,151	2012 June 20	^[16]
3	18	341112⁵²⁴²⁸⁸ + 1	F₁₉(341112)	2,900,832	2012 June 15	^[17]
4	21	75898⁵²⁴²⁸⁸ + 1	F₁₉(75898)	2,558,647	2011 November 19	^[18]
5	42	773620²⁶²¹⁴⁴ + 1	F₁₈(773620)	1,543,643	2012 April 19	^[19]
6	45	676754²⁶²¹⁴⁴ + 1	F₁₈(676754)	1,528,413	2012 February 12	^[20]
7	48	525094²⁶²¹⁴⁴ + 1	F₁₈(525094)	1,499,526	2012 January 18	^[21]
8	52	361658²⁶²¹⁴⁴ + 1	F₁₈(361658)	1,457,075	2011 October 29	^[22]
9	62	145310²⁶²¹⁴⁴ + 1	F₁₈(145310)	1,353,265	2011 February 8	^[23]
10	74	40734²⁶²¹⁴⁴ + 1	F₁₈(40734)	1,208,473	2011 March 8	^[24]

Notes

↑ Křížek, Luca & Somer 2001, p. 38, Remark 4.15
↑ Chris Caldwell, "Prime Links++: special forms" at The Prime Pages.
↑ Ribenboim 1996, p. 88.
1 2 3 4 Keller, Wilfrid (February 7, 2012), "Prime Factors of Fermat Numbers", ProthSearch.net, retrieved September 14, 2016
↑ "PrimeGrid's Mega Prime Search – 193*2^3329782+1 (official announcement)" (PDF). PrimeGrid. Retrieved 7 August 2014.
↑ FermatSearch.org
↑ https://books.google.com/books?id=hgfSBwAAQBAJ&lpg=PA132&ots=On_KUoiNZp&dq=cipolla%20fermat%201904&pg=PA132#v=onepage&q&f=false
↑ Prime Numbers: A Computational Perspective, Richard Crandall and Carl Pomerance, Second edition, Springer, 2011 Look up Selfridge's Conjecture in the Index.
↑ Jeppe Stig Nielsen, "S(n) = n^n + 1".
↑ Generalized Fermat primes for bases up to 1000
↑ Generalized Fermat primes for bases up to 1030
↑ Generalized Fermat primes in odd bases
↑ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
↑ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
↑ "PrimeGrid's Generalized Fermat Prime Search - 475856^524288+1" (PDF). Primegrid. Retrieved 21 August 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 356926^524288+1" (PDF). Primegrid. Retrieved 30 July 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 341112^524288+1" (PDF). Primegrid. Retrieved 9 July 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 75898^524288+1" (PDF). Primegrid. Retrieved 9 July 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 773620^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 676754^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 525094^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 361658^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 145310^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
↑ "PrimeGrid's Generalized Fermat Prime Search - 40734^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.

References

Golomb, S. W. (January 1, 1963), "On the sum of the reciprocals of the Fermat numbers and related irrationalities" (PDF), Canadian Journal of Mathematics, Canadian Mathematical Society, 15: 475–478, doi:10.4153/CJM-1963-051-0, ISSN 0008-414X
Grytczuk, A.; Luca, F. & Wójtowicz, M. (2001), "Another note on the greatest prime factors of Fermat numbers", Southeast Asian Bulletin of Mathematics, Springer-Verlag, 25 (1): 111–115, doi:10.1007/s10012-001-0111-4, ISSN 0129-2021
Guy, Richard K. (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics, 1 (3rd ed.), New York: Springer Verlag, pp. A3, A12, B21, ISBN 0-387-20860-7
Křížek, Michal; Luca, Florian & Somer, Lawrence (2001), 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS books in mathematics, 10, New York: Springer, ISBN 0-387-95332-9 - This book contains an extensive list of references.
Křížek, Michal; Luca, Florian & Somer, Lawrence (2002), "On the convergence of series of reciprocals of primes related to the Fermat numbers" (PDF), Journal of Number Theory, Elsevier, 97 (1): 95–112, doi:10.1006/jnth.2002.2782, ISSN 0022-314X
Luca, Florian (2000), "The anti-social Fermat number", American Mathematical Monthly, Mathematical Association of America, 107 (2): 171–173, doi:10.2307/2589441, ISSN 0002-9890
Ribenboim, Paulo (1996), The New Book of Prime Number Records (3rd ed.), New York: Springer, ISBN 0-387-94457-5
Robinson, Raphael M. (1954), "Mersenne and Fermat Numbers", Proceedings of the American Mathematical Society, American Mathematical Society, 5 (5): 842–846, doi:10.2307/2031878
Yabuta, M. (2001), "A simple proof of Carmichael's theorem on primitive divisors" (PDF), Fibonacci Quarterly, Fibonacci Association, 39: 439–443, ISSN 0015-0517

External links

Fermat prime at Encyclopædia Britannica
Chris Caldwell, The Prime Glossary: Fermat number at The Prime Pages.
Luigi Morelli, History of Fermat Numbers
John Cosgrave, Unification of Mersenne and Fermat Numbers
Wilfrid Keller, Prime Factors of Fermat Numbers
Weisstein, Eric W. "Fermat Number". MathWorld.

Weisstein, Eric W. "Fermat Prime". MathWorld.

Weisstein, Eric W. "Fermat Pseudoprime". MathWorld.

Weisstein, Eric W. "Generalized Fermat Number". MathWorld.

Yves Gallot, Generalized Fermat Prime Search
Mark S. Manasse, Complete factorization of the ninth Fermat number (original announcement)
Peyton Hayslette, Largest Known Generalized Fermat Prime Announcement

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229

List of prime numbers

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

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