Fermat number

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 ⋯ F n − 2 F n = F n − 1 2 − 2 ( F n − 2 − 1 ) 2 F n = F 0 ⋯ 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 ⋯ 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 × 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. 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. 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)

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. 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.

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

∑ n = 0 ∞ 1 ln ⁡ F n = 1 ln ⁡ 2 ∑ n = 0 ∞ 1 log 2 ⁡ ( 2 2 n + 1 ) < 1 ln ⁡ 2 ∑ n = 0 ∞ 2 − n = 2 ln ⁡ 2 .

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

A ∑ n = 0 ∞ ln ⁡ 2 n + 1 ln ⁡ F n = A ∑ n = 0 ∞ log 2 ⁡ 2 n + 1 log 2 ⁡ ( 2 2 n + 1 ) < A ∑ n = 0 ∞ ( n + 1 ) 2 − n = 4 A .

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

C ∑ n = 0 ∞ 2 n + 1 ln ⁡ F n = C ln ⁡ 2 ∑ n = 0 ∞ 2 n + 1 log 2 ⁡ ( 2 2 n + 1 ) > C ln ⁡ 2 ∑ n = 0 ∞ 1 = ∞ ,

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 ≡ − 1 ( mod 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

then N is prime. Conversely, if the above congruence does not hold, and in additionthen 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 × 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:

As of 2016, only F₀ to F₁₁ have been completely factored. The distributed computing project Fermat Search is searching for new factors of Fermat numbers. The set of all Fermat factors is A050922 (or, sorted, A023394) in OEIS.

It is possible that the only primes of this form are 2, 3, 5, 17, 257 and 65,537. Indeed, Boklan and Conway published in 2016 a very precise analysis proving that the probability of the existence of another Fermat prime is inferior to one on a billion.

The following factors of Fermat numbers were known before 1950 (since the 50's digital computers have helped find more factors):

As of September 2016, 331 prime factors of Fermat numbers are known, and 288 Fermat numbers are known to be composite. Several new Fermat factors are found each year.

Pseudoprimes and Fermat numbers

Like composite numbers of the form 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 ≡ 1 ( mod 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 … F s , a > b > ⋯ > s > 1 will be a Fermat pseudoprime to base 2 if and only if 2 s > a .

Other theorems about Fermat numbers

Lemma: If n is a positive integer,

a n − b n = ( a − b ) ∑ k = 0 n − 1 a k b n − 1 − k .

Proof:

( a − b ) ∑ k = 0 n − 1 a k b n − 1 − k = ∑ k = 0 n − 1 a k + 1 b n − 1 − k − ∑ k = 0 n − 1 a k b n − k = a n + ∑ k = 1 n − 1 a k b n − k − ∑ k = 1 n − 1 a k b 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 = r s where 1 ≤ r < k .

By the preceding lemma, for positive integer m ,

( a − b ) ∣ ( a m − b m )

where ∣ means "evenly divides". Substituting a = 2 r , b = − 1 , and m = s and using that s is odd,

( 2 r + 1 ) ∣ ( 2 r s + 1 ) ,

and thus

( 2 r + 1 ) ∣ ( 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 ≡ 1 ( mod p 2 ) does not hold.

It is easy to show 2 m | p − 1 . Now write p − 1 = 2 m λ . If the given congruence holds, then p 2 | 2 2 m λ − 1 , and therefore

0 ≡ ( 2 2 m λ − 1 ) / ( 2 m + 1 ) = ( 2 m − 1 ) ( 1 + 2 2 m + 2 4 m + . . . + 2 2 ( λ − 1 ) m ) ≡ − 2 λ ( mod 2 m + 1 ) .  

Hence 2 m + 1 | 2 λ ,and therefore 2 λ ≥ 2 m + 1 . This leads to

p − 1 ≥ m ( 2 m + 1 ) , which is impossible since m ≥ 2 .

A theorem of Édouard Lucas: Any prime divisor p of F_n = 2 2 n + 1 is of the form k 2 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 n + 1 is the square of 2 2 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 k 2 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 s 2 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 ≡ 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 = 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.

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 = ( A × V j ) mod 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 = 2^2m. The equation nⁿ + 1 = F_(2^m+m) holds at that time.

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

P ( F n ) ≥ 2 n + 2 ( 4 n + 9 ) + 1. (Grytczuk, Luca & Wójtowicz 2001)

Generalized Fermat numbers

Numbers of the form a 2 n + b 2 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 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 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 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 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)

(See for more information (even bases up to 1000), also see for odd bases)

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

(For the smallest even base a such that F n ( a ) is prime, see  A056993)

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 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. They are all megaprimes. As of April 2015 the whole top-10 was discovered by participants in the PrimeGrid project.