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Fermat number

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Fermat number

 Named after Pierre de Fermat 5 5 Fermat numbers 3, 5, 17, 257, 65537 65537 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 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 2015, the only known Fermat primes are F0, F1, F2, F3, and F4 (sequence A019434 in OEIS).

Contents

• Basic properties 1
• Primality of Fermat numbers 2
• Heuristic arguments for density 2.1
• Equivalent conditions of primality 2.2
• Factorization of Fermat numbers 3
• Pseudoprimes and Fermat numbers 4
• Selfridge's Conjecture 5
• Other theorems about Fermat numbers 6
• Relationship to constructible polygons 7
• Applications of Fermat numbers 8
• Pseudorandom Number Generation 8.1
• Other interesting facts 9
• Generalized Fermat numbers 10
• Generalized Fermat primes 10.1
• Largest known generalized Fermat primes 10.2
• Notes 12
• References 13

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 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.

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 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 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:

• Is Fn 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 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

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{align}\sum_{n=0}^{\infty} \frac{1}{\ln F_{n}} &= \frac{1}{\ln 2} \sum_{n=0}^{\infty} \frac{1}{\log_{2}(2^{2^{n}}+1)}\\ &< \frac{1}{\ln 2} \sum_{n=0}^{\infty} 2^{-n} \\ &= \frac{2}{\ln 2}.\end{align}

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 Aln(B)/ln(n), then using Euler's theorem that the least prime factor of Fn exceeds 2n + 1, we would find instead

\begin{align}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}(2^{2^{n}}+1)} \\ &< A \sum_{n=0}^{\infty} (n+1) 2^{-n} \\ &= 4A.\end{align}

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

\begin{align}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}(2^{2^{n}}+1)} \\ &> \frac{C}{\ln 2} \sum_{n=0}^{\infty} 1 \\ &= \infty,\end{align}

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

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 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\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, 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) 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 = 323,170,060,713,110,073,007,148,...,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 2015, 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 April 2015, 322 prime factors of Fermat numbers are known, and 280 Fermat numbers are known to be composite.[4]

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 1964, Rotkiewicz showed that the product of at least two prime or composite Fermat numbers will be a Fermat pseudoprime to base 2.

Selfridge's Conjecture

John L. Selfridge made an intriguing conjecture. Let g(n) be the number of distinct prime factors of 22n + 1 (sequence A046052 in OEIS). Then g(n) is not monotonic (nondecreasing). If another Fermat prime exists, that would imply the conjecture.[7]

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 \le 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)(1+2^{2m}+2^{4m}+...+2^{2(\lambda-1)m}) \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 Fn = 2^{2^{\overset{n}{}}}+1 is of the form k2^{n+2}+1 whenever n is greater than one.

Sketch of proof:

Let Gp 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 Gp (since 2^{2^{\overset{n+1}{}}} is the square of 2^{2^{\overset{n}{}}} which is -1 mod Fn), 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 Gp 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 (1 +2^{2^{n-1}})^{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

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 at that time.[8]

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

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 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 an perfect power with an odd exponent (sequence A070265 in 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 [9][10] for more information (even bases up to 1000), also see [11] for odd bases)

(For the smallest prime of the form F_n(a,b) (with odd a+b), see  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, ... A253854 18 24518, 40734, 145310, 361658, 525094, 676754, 773620, ... A244150 19 75898, 341112, 356926, 475856, ... A243959

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, ... (sequence A056993 in 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 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.[12] They are all megaprimes. As of April 2015 the whole top-10 was discovered by participants in the PrimeGrid project.

Rank Prime rank[13] Prime number Generalized Fermat notation Number of digits Found date reference
1 16 475856524288 + 1 F19(475856) 2,976,633 2012 August 8 [14]
2 17 356926524288 + 1 F19(356926) 2,911,151 2012 June 20 [15]
3 18 341112524288 + 1 F19(341112) 2,900,832 2012 June 15 [16]
4 21 75898524288 + 1 F19(75898) 2,558,647 2011 November 19 [17]
5 42 773620262144 + 1 F18(773620) 1,543,643 2012 April 19 [18]
6 45 676754262144 + 1 F18(676754) 1,528,413 2012 February 12 [19]
7 48 525094262144 + 1 F18(525094) 1,499,526 2012 January 18 [20]
8 52 361658262144 + 1 F18(361658) 1,457,075 2011 October 29 [21]
9 62 145310262144 + 1 F18(145310) 1,353,265 2011 February 8 [22]
10 74 40734262144 + 1 F18(40734) 1,208,473 2011 March 8 [23]

Notes

1. ^ Křížek, Luca & Somer 2001, p. 38, Remark 4.15
2. ^ Chris Caldwell, "Prime Links++: special forms" 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.net, retrieved March 5, 2012
5. ^ "PrimeGrid’s Mega Prime Search – 193*2^3329782+1 (official announcement)" (PDF). PrimeGrid. Retrieved 7 August 2014.
6. ^ FermatSearch.org
7. ^ Prime Numbers: A Computational Perspective, Richard Crandall and Carl Pomerance, Second edition, Springer, 2011 Look up Selfridge's Conjecture in the Index.
8. ^ Jeppe Stig Nielsen, "S(n) = n^n + 1".
9. ^ Generalized Fermat primes for bases up to 1000
10. ^ Generalized Fermat primes for bases up to 1030
11. ^ http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt Generalized Fermat primes in odd bases]
12. ^ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
13. ^ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
14. ^ "PrimeGrid’s Generalized Fermat Prime Search - 475856^524288+1" (PDF). Primegrid. Retrieved 21 August 2012.
15. ^ "PrimeGrid’s Generalized Fermat Prime Search - 356926^524288+1" (PDF). Primegrid. Retrieved 30 July 2012.
16. ^ "PrimeGrid’s Generalized Fermat Prime Search - 341112^524288+1" (PDF). Primegrid. Retrieved 9 July 2012.
17. ^ "PrimeGrid’s Generalized Fermat Prime Search - 75898^524288+1" (PDF). Primegrid. Retrieved 9 July 2012.
18. ^ "PrimeGrid’s Generalized Fermat Prime Search - 773620^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
19. ^ "PrimeGrid’s Generalized Fermat Prime Search - 676754^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
20. ^ "PrimeGrid’s Generalized Fermat Prime Search - 525094^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
21. ^ "PrimeGrid’s Generalized Fermat Prime Search - 361658^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
22. ^ "PrimeGrid’s Generalized Fermat Prime Search - 145310^262144+1" (PDF). Primegrid. Retrieved 9 July 2012.
23. ^ "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,
• 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,
• 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, - 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,
• Luca, Florian (2000), "The anti-social Fermat number", American Mathematical Monthly (Mathematical Association of America) 107 (2): 171–173,
• Robinson, Raphael M. (1954), "Mersenne and Fermat Numbers", Proceedings of the American Mathematical Society (American Mathematical Society) 5 (5): 842–846,
• Yabuta, M. (2001), "A simple proof of Carmichael's theorem on primitive divisors" (PDF), Fibonacci Quarterly (Fibonacci Association) 39: 439–443,