Fermat number

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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 2n + 1 is prime, and n > 0, it can be shown that n must be a power of two. (If n = ab where 1 ≤ a, bn and b is odd, then 2n + 1 = (2a)b + 1 ≡ (−1)b + 1 = 0 (mod 2a + 1). See Sec. 5 for complete proof.) In other words, every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F0, F1, F2, F3, and F4 (sequence A019434 in OEIS).

Basic properties[edit]

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:

  • The number of digits D(n,b) of Fn expressed in the base b is
D(n,b) = \left\lfloor \log_{b}\left(2^{2^{\overset{n}{}}}+1\right)+1 \right\rfloor \approx \lfloor 2^{n}\,\log_{b}2+1 \rfloor (See floor function).
  • No Fermat number can be expressed as the sum of two primes, with the exception of F1 = 2 + 3.
  • 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[edit]

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured (but admitted he could not prove) that all Fermat numbers are prime. Indeed, the first five Fermat numbers F0,...,F4 are easily shown to be prime. However, 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).

It is widely believed that Fermat was 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 and was so convinced of the correctness of his claim that he failed to double-check his work.

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?
  • Are there any Fermat number which is not square-free?

As of 2014 it is known that Fn is composite for 5 ≤ n ≤ 32, although complete factorizations of Fn are known only for 0 ≤ n ≤ 11, and there are no known factors for n = 20 and n = 24.[4] The largest Fermat number known to be composite is F3329780, and its prime factor 193×23329782 + 1, a megaprime, was discovered by the PrimeGrid collaboration in July 2014.[4][5]

Heuristic arguments for density[edit]

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 at most A/ln(n), where A is a fixed constant. Therefore, the total expected number of Fermat primes is at most

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

It should be stressed that this argument is in no way a rigorous proof. For one thing, the argument assumes that Fermat numbers behave "randomly", yet we have already seen that the factors of Fermat numbers have special properties. If (more sophisticatedly) we regard the conditional probability that n is prime, given that we know all its prime factors exceed B, as at most 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[edit]

See also: Pépin's test

There are a number of conditions that are equivalent to the primality of Fn.

  • 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.
  • Let n ≥ 3 be a positive odd integer. Then n is a Fermat prime if and only if for every a co-prime to n, a is a primitive root modulo n if and only if a is a quadratic nonresidue modulo n.
  • The Fermat number Fn > 3 is prime if and only if it can be written uniquely as a sum of two nonzero squares, namely
When F_{n} = x^2 + y^2 not of the form shown above, a proper factor is:
\gcd(x + 2^{2^{n-1}} y, F_{n}).\!
Example 1: F5 = 622642 + 204492, so a proper factor is
\gcd(62264\, +\, 2^{2^4}\times 20449,\, F_{5}) = 641.\!
Example 2: F6 = 40468032562 + 14387937592, so a proper factor is
\gcd(4046803256\, +\, 2^{2^5}\times 1438793759,\, F_{6}) = 274177.\!

Factorization of Fermat numbers[edit]

Because of the size of Fermat numbers, it is difficult to factorize or to prove primality of those. 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; this is in itself almost sufficient 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,
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) ×
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 2014, 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 March 2014, 315 prime factors of Fermat numbers are known, and new ones are discovered each year.

Pseudoprimes and Fermat numbers[edit]

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

2^{F_n-1} \equiv 1 \pmod{F_n}\,\!

for all Fermat numbers.

It is generally believed that all but the first few Fermat numbers are composite. If proven true, this would mean it is possible to generate infinitely many strong pseudoprimes to base 2 from the 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[edit]

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

Other theorems about Fermat numbers[edit]

Lemma: If n is a positive integer,

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



Theorem: If 2^n+1 is an odd prime, then n is a power of 2.


If n is a positive integer but not a power of 2, then n = rs where 1 \le r < n, 1 <  s \le n and r and s are co prime. WLG let s is odd.

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^n+1).

Because 1 < 2^r+1 < 2^n+1, it follows that 2^n+1 is not prime. Therefore, by contraposition n 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, then the congruence 2^{p-1} \equiv 1 \pmod {p^2} does not satisfy.

It is easy to show 2m |p-1. Now write, p-1=2m\lambda. If the given congruence satisfies, then p^2|2^{2m\lambda}-1, 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[edit]

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 = 2kp1p2ps, where k is a nonnegative integer and the pi are distinct Fermat primes.

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

Applications of Fermat numbers[edit]

Pseudorandom Number Generation[edit]

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

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[edit]

A Fermat number cannot be a perfect number or part of a pair of amicable numbers. (Luca 2000)

The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Křížek, Luca & Somer 2002)

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

Numbers of the form a^{2^{ \overset{n} {}}} + b^{2^{ \overset{n} {}}}, where a > 1 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[edit]

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. By analogy with the heuristic argument for the finite number of primes among the base-2 Fermat numbers, it is to be expected that there will be only finitely many generalized Fermat primes for each even base. The smallest prime number Fn(a) with n > 4 is F5(30), or 3032+1.

A more elaborate theory can be used to predict the number of bases for which Fn(a) will be prime for a 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[edit]

The following is a list of the 10 largest known generalized Fermat primes.[9] They are all megaprimes. As of March 2014 the whole top-10 was discovered by participants in the PrimeGrid project.

Rank Largest known prime rank[10] Prime number Generalized Fermat notation Number of digits Found date reference
1 13 475856524288 + 1 F19(475856) 2,976,633 2012 August 8 [11]
2 14 356926524288 + 1 F19(356926) 2,911,151 2012 June 20 [12]
3 15 341112524288 + 1 F19(341112) 2,900,832 2012 June 15 [13]
4 18 75898524288 + 1 F19(75898) 2,558,647 2011 November 19 [14]
5 33 773620262144 + 1 F18(773620) 1,543,643 2012 April 19 [15]
6 35 676754262144 + 1 F18(676754) 1,528,413 2012 February 12 [16]
7 37 525094262144 + 1 F18(525094) 1,499,526 2012 January 18 [17]
8 40 361658262144 + 1 F18(361658) 1,457,075 2011 October 29 [18]
9 43 145310262144 + 1 F18(145310) 1,353,265 2011 February 8 [19]
10 49 40734262144 + 1 F18(40734) 1,208,473 2011 March 8 [20]

See also[edit]


  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 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)". 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. ^ Caldwell, Chris K. "Top Twenty's Generalized Fermat Primes". The Prime Pages. Retrieved 11 March 2014. 
  10. ^ Caldwell, Chris K. "Search on Generalized Fermat". The Prime Pages. Retrieved 11 March 2014. 
  11. ^ "PrimeGrid’s Generalized Fermat Prime Search - 475856^524288+1". Primegrid. Retrieved 21 August 2012. 
  12. ^ "PrimeGrid’s Generalized Fermat Prime Search - 356926^524288+1". Primegrid. Retrieved 30 July 2012. 
  13. ^ "PrimeGrid’s Generalized Fermat Prime Search - 341112^524288+1". Primegrid. Retrieved 9 July 2012. 
  14. ^ "PrimeGrid’s Generalized Fermat Prime Search - 75898^524288+1". Primegrid. Retrieved 9 July 2012. 
  15. ^ "PrimeGrid’s Generalized Fermat Prime Search - 773620^262144+1". Primegrid. Retrieved 9 July 2012. 
  16. ^ "PrimeGrid’s Generalized Fermat Prime Search - 676754^262144+1". Primegrid. Retrieved 9 July 2012. 
  17. ^ "PrimeGrid’s Generalized Fermat Prime Search - 525094^262144+1". Primegrid. Retrieved 9 July 2012. 
  18. ^ "PrimeGrid’s Generalized Fermat Prime Search - 361658^262144+1". Primegrid. Retrieved 9 July 2012. 
  19. ^ "PrimeGrid’s Generalized Fermat Prime Search - 145310^262144+1". Primegrid. Retrieved 9 July 2012. 
  20. ^ "PrimeGrid’s Generalized Fermat Prime Search - 40734^262144+1". Primegrid. Retrieved 9 July 2012. 


External links[edit]