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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^{ \overset{n} {}}} + 1

where n is a nonnegative integer. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The first nine Fermat numbers are (sequence A000215 in OEIS):

F0=21+1=3
F1=22+1=5
F2=24+1=17
F3=28+1=257
F4=216+1=65,537
F5=232+1=4,294,967,297
=641 × 6,700,417
F6=264+1=18,446,744,073,709,551,617
=274,177 × 67,280,421,310,721
F7=2128+1=340,282,366,920,938,463,463,374,607,431,768,211,457
=59,649,589,127,497,217 × 5,704,689,200,685,129,054,721
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
=1,238,926,361,552,897 × 93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321

As of 2007, only the first 12 Fermat numbers have been completely factored. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. These factorizations can be found at Prime Factors of Fermat Numbers

If 2n + 1 is prime, and n > 0, it can be shown that n must be a power of two. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 (If n = ab where 1 <= a, b <= n and b is odd, then 2n + 1 ≡ (2a)b + 1 ≡ (−1)b + 1 ≡ 0 (mod 2a + 1). See below 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,. . . ,F4.

Contents

Basic properties

The Fermat numbers satisfy the following recurrence relations

F_{n} = (F_{n-1}-1)^{2}+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. "Difference equation" redirects here It should not be confused with a Differential equation. Each of these relations can be proved by mathematical induction. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than 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. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield 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. A corollary is a statement which follows readily from a previously proven statement Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness

Further properties:

D(n,b) = \lfloor \log_{b}\left(2^{2^{\overset{n}{}}}+1\right)+1 \rfloor \approx \lfloor 2^{n}\,\log_{b}2+1 \rfloor (See floor function)

Primality of Fermat numbers

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Solomon Wolf Golomb (b 1932 in Baltimore Maryland) is a mathematician and engineer a professor of electrical engineering at the University of Southern California In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of 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 \cdot 6700417. \;

Euler proved that every factor of Fn must have the form k2n+1 + 1. For n = 5, this means that the only possible factors are of the form 64k + 1. Euler found the factor 641 = 10×64 + 1.

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. 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, very little is known about Fermat numbers with large n. [1] In fact, each of the following is an open problem:

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. A composite number is a positive Integer which has a positive Divisor other than one or itself Ferdinand Gotthold Max Eisenstein ( 16 April, 1823 – 11 October, 1852) was a German Mathematician. A heuristic argument is an Argument that reasons from the value of a method or principle that has been shown by experimental (especially Trial-and-error) investigation Probability is the likelihood or chance that something is the case or will happen Therefore, the total expected number of Fermat primes is at most

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}.

It should be stressed that this argument is in no way a rigorous proof. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true 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. Randomness is a lack of order Purpose, cause, or predictability Although it is widely believed that there are only finitely many Fermat primes, there are some experts who disagree. [2]

As of 2008 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 in {14, 20, 22, 24}. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common The largest Fermat number known to be composite is F2478782, and its prime factor 3×22478785 + 1 was discovered by John B. Cosgrave and his Proth-Gallot Group on October 10, 2003. Dr John B Cosgrave (January 5 1946 is the head of the mathematics department of St Events 680 - Battle of Karbala: Shia Imam Husayn bin Ali, the grandson of the Prophet Muhammad, is decapitated Year 2003 ( MMIII) was a Common year starting on Wednesday of the Gregorian calendar. An even more speculative application of the heuristic argument above suggests - subject to the same caveats - that the "probability" that there are any new Fermat primes beyond F32 is on the order of one in a billion. Probability is the likelihood or chance that something is the case or will happen

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

a^{(N-1)/2} \equiv -1 \mod 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. The Jacobi symbol is a generalization of the Legendre symbol introduced by Jacobi in 1837 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. In Mathematics, Pépin's test is a Primality test, which can be used to determine whether a Fermat number is prime. Although Pépin's test and Proth's theorem have been implemented on computers to prove the compositeness of many Fermat numbers, neither test gives a specific nontrivial factor. In fact, no specific prime factors are known for n = 14, 20, 22, and 24.
F_{n}=\left(2^{2^{n-1}}\right)^{2}+1^{2}
When Fn = x2 + y2 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}\, 20449,\, F_{5}) = 641.
Example 2: F6 = 40468032562 + 14387937592, so a proper factor is \gcd(4046803256\, +\, 2^{2^5}\, 1438793759,\, F_{6}) = 274177.

Factorization of Fermat numbers

Because of the size of Fermat numbers, it is difficult to factorize or to prove primality of those. Pépin's test is necessary and sufficient test for primality of Fermat numbers which can be implemented by modern computers. In Mathematics, Pépin's test is a Primality test, which can be used to determine whether a Fermat number is prime. The elliptic curve method is a fast method for finding small prime divisors of numbers. The Lenstra elliptic curve factorization or the elliptic curve factorization method ( ECM) is a fast sub- Exponential running time algorithm for Integer 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. Edouard Lucas proved in 1878 that every factor of Fermat number Fn is of the form 2n + 2k + 1, where k is a positive integer. François Édouard Anatole Lucas ( April 4, 1842 in Amiens - October 3, 1891) was a French Mathematician.

Pseudoprimes and Fermat numbers

Like composite numbers of the form 2p-1, every composite Fermat number is a strong pseudoprime to base 2. A composite number is a positive Integer which has a positive Divisor other than one or itself In Number theory, a strong pseudoprime is a Composite number that passes a pseudoprimality test Because all strong pseudoprimes to be 2 are also Fermat pseudoprimes - ie. A pseudoprime is a Probable prime (an Integer which shares a property common to all Prime numbers which is not actually prime

2^{Fn} \equiv 1 \pmod{Fn}\,\!

for all Fermat numbers.

Because it is generally believed that all but the first few Fermat numbers are composite, this makes it possible to generate infinitely many strong pseudoprimes to base 2 from the Fermat numbers.

In fact, Rotkiewicz showed in 1964 that the product of any number of prime or composite Fermat numbers will be a Fermat pseudoprime to base 2.

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
= an − bn

Theorem: If 2n + 1 is prime, then n is zero or a power of 2.

proof:

For n = 0, 20 + 1 equals prime number 2. (This is why some sources count 2 as a sixth Fermat prime. )

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 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 = 2r, b = − 1, and m = s,

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

and thus

(2^r+1) \mid (2^n+1).

Because 2r + 1 > 1, 2n + 1 is not prime when n is a positive integer that is not a power of 2.

A theorem of Édouard Lucas: Any prime divisor p of Fn = 2^{2^{\overset{n}{}}}+1 is of the form k2n + 2 + 1 whenever n is greater than one. François Édouard Anatole Lucas ( April 4, 1842 in Amiens - October 3, 1891) was a French Mathematician.

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 2n + 1 in Gp, so that, by Lagrange's theorem, p-1 is divisible by 2n + 1 and p has the form k2n + 1 + 1 for some integer k, as Euler knew. Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of 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 in integer a such that a2 -2 is divisible by p. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German An Integer q is called a quadratic residue modulo n if it is congruent to a perfect square (mod n) i Then the image of a has order 2n + 2 in the group Gp and (using Lagrange's theorem again), p-1 is divisible by 2n + 2 and p has the form s2n + 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 a power of 2 or the product of a power of 2 and distinct Fermat primes. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles In other words, if and only if n is of the form n = 2kp1p2. . . ps, where k is a nonnegative integer and the pi are distinct Fermat primes. See constructible polygon. In mathematics a constructible polygon is a Regular polygon that can be constructed with compass and straightedge.

A positive integer n is of the above form if and only if φ(n) is a power of 2, where φ(n) is Euler's totient function. In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to

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 relatively prime to P. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than Then take the result MOD 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. A linear congruential generator ( LCG) represent one of the oldest and best-known Pseudorandom number generator Algorithms The theory behind them is easy RANDU is an infamous linear congruential Pseudorandom number generator which has been used since the 1960s 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 pseudorandom process is a process that appears random but is not 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. Amicable numbers are two different Numbers so related that the sum of the Proper divisors of the one is equal to the other one being considered (Luca 2000)

The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Krizek, 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. [3]

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

P(F_n )\ge 2^{m+2}(4m+9)+1. (Grytczuk, Luca and Wojtowicz, 2001）

A Fermat prime cannot also be a Wieferich prime. (Luca)

Generalized Fermat numbers

Numbers of the form a^{2^{ \overset{n} {}}} + b^{2^{ \overset{n} {}}}, where a>1 are called generalized Fermat numbers. 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)

An odd prime p is a generalized Fermat number if and only if p is congruent to 1 ( mod 4).

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. 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. A heuristic argument is an Argument that reasons from the value of a method or principle that has been shown by experimental (especially Trial-and-error) investigation 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.

References

See also

External links

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