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Prime number

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Divisibility-based
   sets of integers
   Form of factorization:
   Prime number
   Composite number
   Powerful number
   Square-free number
   Achilles number
   Constrained divisor sums:
   Perfect number
   Almost perfect number
   Quasiperfect number
   Multiply perfect number
   Hyperperfect number
   Unitary perfect number
   Semiperfect number
   Primitive semiperfect number
   Practical number
   Numbers with many divisors:
   Abundant number
   Highly abundant number
   Superabundant number
   Colossally abundant number
   Highly composite number
   Superior highly composite number
   Other:
   Deficient number
   Weird number
   Amicable number
   Sociable number
   Sublime number
   Harmonic divisor number
   Frugal number
   Equidigital number
   Extravagant number
   See also:
   Divisor function
   Divisor
   Prime factor
   Factorization

   In mathematics, a prime number (or a prime) is a natural number that
   has exactly two (distinct) natural number divisors, which are 1 and the
   prime number itself. There exists an infinitude of prime numbers, as
   demonstrated by Euclid, in about 300 B.C.. The first 30 prime numbers
   are 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, and 113 (sequence
   A000040 in OEIS); see the list of prime numbers for a longer list.

   The property of being a prime is called primality, and the word prime
   is also used as an adjective. Since 2 is the only even prime number,
   the term odd prime refers to all prime numbers greater than 2.

   The study of prime numbers is part of number theory, the branch of
   mathematics which encompasses the study of natural numbers. Prime
   numbers have been the subject of intense research, yet some fundamental
   questions, such as the Riemann hypothesis and the Goldbach conjecture,
   have been unresolved for more than a century. The problem of modelling
   the distribution of prime numbers is a popular subject of investigation
   for number theorists: when looking at individual numbers, the primes
   seem to be randomly distributed, but the "global" distribution of
   primes follows well-defined laws.

   The notion of prime number has been generalized in many different
   branches of mathematics.
     * In ring theory, a branch of abstract algebra, the term "prime
       element" has a specific meaning. Here, a non-zero, non-unit ring
       element a is defined to be prime if whenever a divides b c for ring
       elements b and c, then a divides at least one of b or c. With this
       meaning, the additive inverse of any prime number is also prime. In
       other words, when considering the set of integers ( \mathbb{Z} ) as
       a ring, − 7 is a prime element. Without further specification,
       however, "prime number" always means a positive integer prime.
       Among rings of complex algebraic integers, Eisenstein primes, and
       Gaussian primes may also be of interest.
     * In knot theory, a prime knot is a knot which can not be
       disaggregated into a smaller prime knot.

   In both the two above examples, the fundamental theorem of arithmetic
   (Every natural number can be 'uniquely' decomposed into a product of
   primes) does not apply.

History of prime numbers

   There are hints in the surviving records of the ancient Egyptians that
   they had some knowledge of prime numbers: the Egyptian fraction
   expansions in the Rhind papyrus, for instance, have quite different
   forms for primes and for composites. However, the earliest surviving
   records of the explicit study of prime numbers come from the Ancient
   Greeks. Euclid's Elements (circa 300 BC) contain important theorems
   about primes, including the infinitude of primes and the fundamental
   theorem of arithmetic. Euclid also showed how to construct a perfect
   number from a Mersenne prime. The Sieve of Eratosthenes, attributed to
   Eratosthenes, is a simple, efficient and still widely used method to
   compute primes.

   After the Greeks, little happened until the 17th century. In 1640
   Pierre de Fermat stated (without proof) Fermat's little theorem (later
   proved by Leibnitz and Euler). A special case of it may have been known
   long before by the Chinese. Other conjectures by Fermat turned out to
   be false, e.g. that all Fermat numbers are prime. The French monk Marin
   Mersenne made partially false assertions about Mersenne primes.

   In 1747 Euler showed that all even perfect numbers are on Euclid's
   form. It is still unknown whether there exist odd perfect numbers.
   Euler's extensive work included many significant results about primes,
   e.g. that the infinite sum 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... is
   divergent.

   Several mathematicians made contributions towards the prime number
   theorem until it was proved independently by Hadamard and de la Vallée
   Poussin in 1896.

   Fermat's little theorem does not prove primality which can be
   determined by slow trial division for small numbers. Many
   mathematicians have worked on faster primality tests for large numbers,
   often restricted to specific number forms. This includes Pépin's test
   for Fermat numbers (1877), Proth's theorem (around 1878), the
   Lucas–Lehmer test for Mersenne numbers (originated 1878), and the
   generalized Lucas–Lehmer test. More recent algorithms like APRT-CL,
   ECPP and AKS work on arbitrary numbers but remain much slower.

   Until the 19th century most mathematicians considered the number 1 a
   prime, and there is still a large body of mathematical work that is
   valid despite labelling 1 a prime, such as the work of Stern and
   Zeisel. Henri Lebesgue is said to be the last professional
   mathematician to call 1 prime. The change in label occurred so that it
   can be said 'each number has a unique factorization into primes.' See
   also "Arguments for and against the primality of 1".

   For a long time, prime numbers were thought as having no possible
   application outside of number theory; this changed in the 1970s when
   the concepts of public-key cryptography were invented, in which prime
   numbers formed the basis of the first algorithms such as the RSA
   cryptosystem or the Diffie-Hellman key-exchange algorithm.

   Since 1951 all the largest known primes have been found by computers.
   The search for ever larger primes has generated interest outside
   mathematical circles. Great Internet Mersenne Prime Search and other
   distributed computing projects to find large primes have become
   popular, while mathematicians continue to struggle with the theory of
   primes.

Prime divisors

   The fundamental theorem of arithmetic states that every positive
   integer larger than 1 can be written as a product of one or more primes
   in a unique way, i.e. unique except for the order. The same prime may
   occur multiple times. Primes can thus be considered the "basic building
   blocks" of the natural numbers. For example, we can write

          23244 = 2^2 \times 3 \times 13 \times 149

   and any other factorization of 23244 as the product of primes will be
   identical except for the order of the factors. There are many prime
   factorization algorithms to do this in practice for larger numbers.

   The importance of this theorem is one of the reasons for the exclusion
   of 1 from the set of prime numbers. If 1 were admitted as a prime, the
   precise statement of the theorem would require additional
   qualifications.

Properties of primes

     * The base 10 numerals for all prime numbers except 2 and 5 end in 1,
       3, 7, or 9. (Numerals ending in 0, 2, 4, 6 or 8 represent multiples
       of 2 and numerals ending in 5 represent multiples of 5, due to the
       use of the base 10 (2 × 5) system.)

     * If p is a prime number and p divides a product ab of integers, then
       p divides a or p divides b. This proposition was proved by Euclid
       and is known as Euclid's lemma. It is used in some proofs of the
       uniqueness of prime factorizations.

     * The ring Z/nZ (see modular arithmetic) is a field if and only if n
       is a prime. Put another way: n is prime if and only if φ(n) = n −
       1.

     * If p is prime and a is any integer, then a^p − a is divisible by p
       ( Fermat's little theorem).

     * If p is a prime number other than 2 and 5, 1/p is always a
       recurring decimal, with a period of p-1 or a divisor of p-1. This
       can be deduced directly from Fermat's little theorem. 1/p expressed
       likewise in base q (i.e. other than base 10) has similar effect,
       provided that p is not a prime factor of q. The article on
       recurring decimals shows some of the interesting properties.

     * An integer p > 1 is prime if and only if the factorial (p − 1)! + 1
       is divisible by p ( Wilson's theorem). Conversely, an integer n > 4
       is composite if and only if (n − 1)! is divisible by n.

     * If n is a positive integer greater than 1, then there is always a
       prime number p with n < p < 2n ( Bertrand's postulate).

     * Adding the reciprocals of all primes together results in a
       divergent infinite series ( proof). More precisely, if S(x) denotes
       the sum of the reciprocals of all prime numbers p with p ≤ x, then
       S(x) = ln ln x + O(1) for x → ∞ (see Big O notation).

     * In every arithmetic progression a, a + q, a + 2q, a + 3q,... where
       the positive integers a and q ≥ 1 are coprime, there are infinitely
       many primes ( Dirichlet's theorem on arithmetic progressions).

     * The characteristic of every field is either zero or a prime number.

     * If G is a finite group and p^n is the highest power of the prime p
       which divides the order of G, then G has a subgroup of order p^n. (
       Sylow theorems)

     * If p is prime and G is a group with p^n elements, then G contains
       an element of order p.

     * The prime number theorem says that the proportion of primes less
       than x is asymptotic to 1/ln x (in other words, as x gets very
       large, the likelihood that a number less than x is prime is
       inversely proportional to the number of digits in x).

     * The Copeland-Erdős constant 0.235711131719232931374143..., obtained
       by concatenating the prime numbers, is known to be an irrational
       number.

     * The value of the Riemann zeta function at each point in the complex
       plane is given as a meromorphic continuation of a function, defined
       by a product over the set of all primes for Re(s)>1:

                \zeta(s)= \sum_{n=1}^\infin \frac{1}{n^s} = \prod_{p}
                \frac{1}{1-p^{-s}}.

          Evaluating this identity at different integers provides an
          infinite number of products over the primes whose values can be
          calculated, the first two being

                \prod_{p} \frac{1}{1-p^{-1}} = \infty
                \prod_{p} \frac{1}{1-p^{-2}}= \frac{\pi^2}{6}.

     * If p>1\; , the polynomial x^{p-1}+x^{p-2}+ \cdots + 1 is
       irreducible over Z / pZ if and only if p\; is prime.

     * If p is a prime number greater than 6 then p mod 6 is either 1 or 5
       and p mod 30 is 1, 7, 11, 13, 17, 19, 23 or 29.

Classification of prime numbers

   Two ways of classifying prime numbers, class n+ and class n-, were
   studied by Paul Erdős and John Selfridge.

   Determining the class n+ of a prime number p involves looking at the
   largest prime factor of p + 1. If that largest prime factor is 2 or 3,
   then p is class 1+. But if that largest prime factor is another prime
   q, then the class n+ of p is one more than the class n+ of q. Sequences
   A005105 through A005108 list class 1+ through class 4+ primes.

   The class n- is almost the same as class n+, except that the
   factorization of p - 1 is looked at instead.

The number of prime numbers

There are infinitely many prime numbers

   The oldest known proof for the statement that there are infinitely many
   prime numbers is given by the Greek mathematician Euclid in his
   Elements (Book IX, Proposition 20). Euclid states the result as "there
   are more than any given [finite] number of primes", and his proof is
   essentially the following:

          Suppose you have a finite number of primes. Call this number m.
          Multiply all m primes together and add one (see Euclid number).
          The resulting number is not divisible by any of the finite set
          of primes, because dividing by any of these would give a
          remainder of one. And one is not divisible by any primes.
          Therefore it must either be prime itself, or be divisible by
          some other prime that was not included in the finite set. Either
          way, there must be at least m + 1 primes. But this argument
          applies no matter what m is; it applies to m + 1, too. So there
          are more primes than any given finite number.

   This previous argument explains why the product of m primes plus 1 must
   be divisible by some prime not among the m primes, or be prime itself.
   A common mistake is thinking Euclid's proof says the prime product plus
   1 is always prime.
   (2 · 3 · 5 · 7 · 11 · 13) + 1 = 30,031 = 59 · 509 (both primes) shows
   this is not the case.

   Other mathematicians have given their own proofs. One of those (due to
   Euler) shows that the sum of the reciprocals of all prime numbers
   diverges to infinity. Kummer's is particularly elegant and Harry
   Furstenberg provides one using general topology.

Counting the number of prime numbers below a given number

   Even though the total number of primes is infinite, one could still ask
   "Approximately how many primes are there below 100,000?", or "How
   likely is a random 20-digit number to be prime?".

   The prime counting function π(x) is defined as the number of primes up
   to x. There are known algorithms to compute exact values of π(x) faster
   than it would be to compute each prime up to x. Values as large as
   π(10^20) can be calculated quickly and accurately with modern
   computers. Thus, e.g., π(100000) = 9592, and π(10^20) =
   2,220,819,602,560,918,840.

   For larger values of x, beyond the reach of modern equipment, the prime
   number theorem provides a good estimate: π(x) is approximately x/ln(x).
   Even better estimates are known.

Location of prime numbers

Finding prime numbers

   The Sieve of Eratosthenes is a simple way, and the Sieve of Atkin a
   fast way, to compute the list of all prime numbers up to a given limit.

   In practice, though, one usually wants to check whether a given number
   is prime, rather than generate a list of primes. Further, it is often
   satisfactory to know the answer with a high probability. It is possible
   to quickly check whether a given large number (say, up to a few
   thousand digits) is prime using probabilistic primality tests. These
   typically pick a random number called a "witness" and check some
   formula involving the witness and the potential prime N. After several
   iterations, they declare N to be "definitely composite" or "probably
   prime". Some of these tests are not perfect: there may be some
   composite numbers, called pseudoprimes for the respective test, that
   will be declared "probably prime" no matter what witness is chosen.
   However, the most popular probabilistic tests do not suffer from this
   drawback.

   One method for determining whether a number is prime is to divide by
   all primes less than or equal to the square root of that number. If any
   of the divisions come out as an integer, then the original number is
   not a prime. Otherwise, it is a prime. One need not actually calculate
   the square root; once one sees that the quotient is less than the
   divisor, one can stop. This is known as trial division; it is the
   simplest primality test and it quickly becomes impractical for testing
   large integers because the number of possible factors grows
   exponentially as the number of digits in the number-to-be-tested
   increases.

Pseudocode for programs to find primes

   The least efficient algorithm starts out with a list containing just
   the number 2, then tries dividing each subsequent number by the primes
   already in the list. If it is not divisible evenly by any of them, it
   is added to the list. But if it is divisible evenly by any number
   already on the list, the program moves on to the next candidate.

   The simplest, most elegant algorithm is perhaps the sieve of
   Eratosthenes.
// arbitrary search limit
limit ← 1000000

// assume all numbers are prime at first
is_prime(i) ← true, i ∈ [2, limit]

for n in [2, √limit]:
    if is_prime(n):
        // eliminate multiples of each prime,
        // starting with its square
        // 2n, 3n, ..., (n-1)n already eliminated
        // nn, (n+1)n, (n+2)n, ... to be eliminated
        is_prime(i) ← false, i ∈ {n², n²+n, n²+2n, ..., limit}

for n in [2, limit]:
    if is_prime(n): print n


   A more complicated, but more efficient algorithm (when properly
   optimized) is the sieve of Atkin. Its basic structure is as follows.
// arbitrary search limit
limit ← 1000000

// initialize the sieve
is_prime(i) ← false, i ∈ [5, limit]

// put in candidate primes:
// integers which have an odd number of
// representations by certain quadratic forms
for (x, y) in [1, √limit] × [1, √limit]:
    n ← 4x²+y²
    if (n ≤ limit) ∧ (n mod 12 = 1 ∨ n mod 12 = 5):
        is_prime(n) ← ¬is_prime(n)
    n ← 3x²+y²
    if (n ≤ limit) ∧ (n mod 12 = 7):
        is_prime(n) ← ¬is_prime(n)
    n ← 3x²-y²
    if (x > y) ∧ (n ≤ limit) ∧ (n mod 12 = 11):
        is_prime(n) ← ¬is_prime(n)

// eliminate composites by sieving
for n in [5, √limit]:
    if is_prime(n):
        // n is prime, omit multiples of its square; this is sufficient because
        // composites which managed to get on the list cannot be square-free
        is_prime(k) ← false, k ∈ {n², 2n², 3n², ..., limit}

print 2, 3
for n in [5, limit]:
    if is_prime(n): print n

Primality tests

   A primality test algorithm is an algorithm which tests a number for
   primality, i.e. whether the number is a prime number.
     * AKS primality test
     * Fermat primality test
     * Lucas-Lehmer test
     * Solovay-Strassen primality test
     * Miller-Rabin primality test

   A probable prime is an integer which, by virtue of having passed a
   certain test, is considered to be probably prime. Probable primes which
   are in fact composite (such as Carmichael numbers) are called
   pseudoprimes.

   In 2002, Indian scientists at IIT Kanpur discovered a new deterministic
   algorithm known as the AKS algorithm. The amount of time that this
   algorithm takes to check whether a number N is prime depends on a
   polynomial function of the number of digits of N (i.e. of the logarithm
   of N).

Formulas yielding prime numbers

   There is no known formula for primes which is more efficient at finding
   primes than the methods mentioned above under "Finding prime numbers".

   There is a set of Diophantine equations in 9 variables and one
   parameter with the following property: the parameter is prime if and
   only if the resulting system of equations has a solution over the
   natural numbers. This can be used to obtain a single formula with the
   property that all its positive values are prime.

   There is no polynomial, even in several variables, that takes only
   prime values. For example, the curious polynomial in one variable
   f(n) = n^2 − n + 41 yields primes for n = 0,..., 40, but f(41) is
   composite. However, there are polynomials in several variables, whose
   positive values as the variables take all positive integer values are
   exactly the primes.

   Another formula is based on Wilson's theorem mentioned above, and
   generates the number two many times and all other primes exactly once.
   There are other similar formulae which also produce primes.

Special types of primes from formulas for primes

   A prime p is called primorial or prime-factorial if it has the form p =
   n# ± 1 for some number n, where n# stands for the product 2 · 3 · 5 · 7
   · 11 · ... of all the primes ≤ n. A prime is called factorial if it is
   of the form n! ± 1. The first factorial primes are:

          n! − 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94,
          166,... (sequence A002982 in OEIS)

          n! + 1 is prime for n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116,
          154... (sequence A002981 in OEIS)

   The largest known primorial prime is Π(392113) + 1, found by Heuer in
   2001. The largest known factorial prime is 34790! − 1, found by
   Marchal, Carmody and Kuosa in 2002. It is not known whether there are
   infinitely many primorial or factorial primes.

   Primes of the form 2^n − 1 are known as Mersenne primes, while primes
   of the form 2^{2^n} + 1 are known as Fermat primes. Prime numbers p
   where 2p + 1 is also prime are known as Sophie Germain primes. The
   following list is of other special types of prime numbers that come
   from formulas:
     * Wieferich primes,
     * Wilson primes,
     * Wall-Sun-Sun primes,
     * Wolstenholme primes,
     * Unique primes,
     * Newman-Shanks-Williams primes (NSW primes),
     * Smarandache-Wellin primes,
     * Wagstaff primes, and
     * Supersingular primes.

   Some primes are classified according to the properties of their digits
   in decimal or other bases. For example, numbers whose digits form a
   palindromic sequence are called palindromic primes, and a prime number
   is called a truncatable prime if successively removing the first digit
   at the left or the right yields only new prime numbers.

The distribution of the prime numbers

   The distribution of all the prime numbers in the range of 1 to 76,800.
   Each pixel represents a number with black pixels meaning that number is
   prime and white ones not-prime.
   The distribution of all the prime numbers in the range of 1 to 76,800.
   Each pixel represents a number with black pixels meaning that number is
   prime and white ones not-prime.

   The problem of modelling the distribution of prime numbers is a popular
   subject of investigation for number theorists. The prime numbers are
   distributed among the natural numbers in a (so far) unpredictable way,
   but there do appear to be laws governing their behaviour. Leonhard
   Euler commented

          Mathematicians have tried in vain to this day to discover some
          order in the sequence of prime numbers, and we have reason to
          believe that it is a mystery into which the mind will never
          penetrate. (Havil 2003, p. 163)

   Paul Erdős said

          God may not play dice with the universe, but something strange
          is going on with the prime numbers. [Referring to Albert
          Einstein's famous belief that "God does not play dice with the
          universe."]

   In a 1975 lecture, Don Zagier commented

          "There are two facts about the distribution of prime numbers of
          which I hope to convince you so overwhelmingly that they will be
          permanently engraved in your hearts. The first is that, despite
          their simple definition and role as the building blocks of the
          natural numbers, the prime numbers grow like weeds among the
          natural numbers, seeming to obey no other law than that of
          chance, and nobody can predict where the next one will sprout.
          The second fact is even more astonishing, for it states just the
          opposite: that the prime numbers exhibit stunning regularity,
          that there are laws governing their behaviour, and that they
          obey these laws with almost military precision." (Havil 2003, p.
          171)

Gaps between primes

   Let p[n] denote the nth prime number (i.e. p[1] = 2, p[2] = 3, etc.).
   The gap g[n] between the consecutive primes p[n] and p[n + 1] is the
   difference between them, i.e.

          g[n] = p[n + 1] − p[n].

   We have g[1] = 3 − 2 = 1, g[2] = 5 − 3 = 2, g[3] = 7 − 5 = 2, g[4] = 11
   − 7 = 4, and so on. The sequence (g[n]) of prime gaps has been
   extensively studied.

   For any natural number N larger than 1, the sequence (for the notation
   N! read factorial)

          N! + 2, N! + 3, ..., N! + N

   is a sequence of N − 1 consecutive composite integers. Therefore, there
   exist gaps between primes which are arbitrarily large, i.e. for any
   natural number N, there is an integer n with g[n] > N. (Choose n so
   that p[n] is the greatest prime number less than N! + 2.)

   On the other hand, the gaps get arbitrarily small in proportion to the
   primes: the quotient g[n]/p[n] approaches zero as n approaches
   infinity. Note also that the twin prime conjecture asserts that g[n] =
   2 for infinitely many integers n.

Location of the largest known prime

   The largest known prime, as of September 2006, is 2^32,582,657 − 1
   (this number is 9,808,358 digits long); it is the 44th known Mersenne
   prime. M[32582657] was found on September 4, 2006 by Curtis Cooper and
   Steven Boone, professors at the University of Central Missouri
   (formerly Central Missouri State University) and members of a
   collaborative effort known as GIMPS. Before finding the prime, Cooper
   and Boone ran the GIMPS program on a peak of 700 CMSU computers for 9
   years.

   The next two largest known primes are also Mersenne Primes:
   M[30,402,457] = 2^30,402,457 − 1 (43rd known Mersenne prime, 9,152,052
   digits long) and M[25964951] = 2^25,964,951 − 1 (42nd known Mersenne
   prime, 7,816,230 digits long). Historically, the largest known prime
   has almost always been a Mersenne prime since the dawn of electronic
   computers, because there exists a particularly fast primality test for
   numbers of this form, the Lucas–Lehmer test for Mersenne numbers.

   The largest known prime that is not a Mersenne prime is 27,653 ×
   2^9,167,433 + 1 (2,759,677 digits), a Proth number. This is also the
   seventh largest known prime of any form. It was found by the Seventeen
   or Bust project and it brings them one step closer to solving the
   Sierpinski problem.

   Some of the largest primes not known to have any particular form (that
   is, no simple formula such as that of Mersenne primes) have been found
   by taking a piece of semi-random binary data, converting it to a number
   n, multiplying it by 256^k for some positive integer k, and searching
   for possible primes within the interval [256^kn + 1, 256^k(n + 1) − 1].

Awards for finding primes

   The Electronic Frontier Foundation (EFF) has offered a $100,000 (U.S.)
   prize to the first discoverers of a prime with at least 10 million
   digits. They also offer $150,000 for 100 million digits, and $250,000
   for 1 billion digits. In 2000 they paid out $50,000 for 1 million
   digits.

   The RSA Factoring Challenge offers prizes up to $200,000 (U.S) for
   finding the prime factors of certain semiprimes of up to 2048 bits.

Generalizations of the prime concept

   The concept of prime number is so important that it has been
   generalized in different ways in various branches of mathematics.

Prime elements in rings

   One can define prime elements and irreducible elements in any integral
   domain. For any unique factorization domain, such as the ring Z of
   integers, the set of prime elements equals the set of irreducible
   elements, which for Z is {..., −11, −7, −5, −3, −2, 2, 3, 5, 7, 11,
   ...}.

   As an example, we consider the Gaussian integers Z[i], that is, complex
   numbers of the form a + bi with a and b in Z. This is an integral
   domain, and its prime elements are the Gaussian primes. Note that 2 is
   not a Gaussian prime, because it factors into the product of the two
   Gaussian primes (1 + i) and (1 − i). The element 3, however, remains
   prime in the Gaussian integers. In general, rational primes (i.e. prime
   elements in the ring Z of integers) of the form 4k + 3 are Gaussian
   primes, whereas rational primes of the form 4k + 1 are not.

Prime ideals

   In ring theory, one generally replaces the notion of number with that
   of ideal. Prime ideals are an important tool and object of study in
   commutative algebra, algebraic number theory and algebraic geometry.
   The prime ideals of the ring of integers are the ideals (0), (2), (3),
   (5), (7), (11), ...

   A central problem in algebraic number theory is how a prime ideal
   factors when it is lifted to an extension field. For example, in the
   Gaussian integer example above, (2) ramifies into a prime power (1 + i
   and 1 − i generate the same prime ideal), prime ideals of the form (4k
   + 3) are inert (remain prime), and prime ideals of the form (4k + 1)
   split (are the product of 2 distinct prime ideals).

Primes in valuation theory

   In class field theory yet another generalization is used. Given an
   arbitrary field K, one considers valuations on K, certain functions
   from K to the real numbers R. Every such valuation yields a topology on
   K, and two valuations are called equivalent if they yield the same
   topology. A prime of K (sometimes called a place of K) is an
   equivalence class of valuations. With this definition, the primes of
   the field Q of rational numbers are represented by the standard
   absolute value function (known as the "infinite prime") as well as by
   the p-adic valuations on Q, for every prime number p.

Prime knots

   In knot theory, a prime knot is a knot which is, in a certain sense,
   indecomposable. Specifically, it is one which cannot be written as the
   knot sum of two nontrivial knots.

Open questions

   There are many open questions about prime numbers. A very significant
   one is the Riemann hypothesis, which essentially says that the primes
   are as regularly distributed as possible. From a physical viewpoint, it
   roughly states that the irregularity in the distribution of primes only
   comes from random noise. From a mathematical viewpoint, it roughly
   states that the asymptotic distribution of primes (about 1/ log x of
   numbers less than x are primes, the prime number theorem) also holds
   for much shorter intervals of length about the square root of x (for
   intervals near x). This hypothesis is generally believed to be correct,
   in particular, the simplest assumption is that primes should have no
   significant irregularities without good reason.

   Many famous conjectures appear to have a very high probability of being
   true (in a formal sense, many of them follow from simple heuristic
   probabilistic arguments):
     * Prime Euclid numbers: It is not known whether or not there are an
       infinite number of prime Euclid numbers.

     * Strong Goldbach conjecture: Every even integer greater than 2 can
       be written as a sum of two primes.

     * Weak Goldbach conjecture: Every odd integer greater than 5 can be
       written as a sum of three primes.

     * Twin prime conjecture: There are infinitely many twin primes, pairs
       of primes with difference 2.

     * Polignac's conjecture: For every positive integer n, there are
       infinitely many pairs of consecutive primes which differ by 2n.
       When n = 1 this is the twin prime conjecture.

     * A weaker form of Polignac's conjecture: Every even number is the
       difference of two primes.

     * It is widely believed there are infinitely many Mersenne primes,
       but not Fermat primes.

     * Many believe there are infinitely many primes of the form n^2 + 1.

     * Many well-known conjectures are special cases of the broad
       Schinzel's hypothesis H.

     * Many believe there are infinitely many Fibonacci primes.

     * Legendre's conjecture: There is a prime number between n^2 and
       (n + 1)^2 for every positive integer n.

     * Cramér's conjecture: \limsup_{n\rightarrow\infty}
       \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1 . This conjecture implies
       Legendre's, but its status is more unsure.

     * Brocard's conjecture: There are always at least four primes between
       the squares of successive primes > 2.

Applications

   For a long time, number theory in general, and the study of prime
   numbers in particular, was seen as the canonical example of pure
   mathematics, with no applications outside of the self-interest of
   studying the topic. In particular, number theorists such as British
   mathematician G. H. Hardy prided themselves on doing work that had
   absolutely no military significance. However, this vision was shattered
   in the 1970s, when it was publicly announced that prime numbers could
   be used as the basis for the creation of public key cryptography
   algorithms. Prime numbers are also used for hash tables and
   pseudorandom number generators.

   Some rotor machines were designed with a different number of pins on
   each rotor, with the number of pins on any one rotor either prime, or
   coprime to the number of pins on any other rotor. This helped generate
   the full cycle of possible rotor positions before repeating any
   position.

Public-key cryptography

   Several public-key cryptography algorithms, such as RSA, are based on
   large prime numbers (that is, greater than 10^100).

Prime numbers in nature

   Many numbers occur in nature, and inevitably some of these are prime.
   There are, however, relatively few examples of numbers that appear in
   nature because they are prime. For example, most starfish have 5 arms,
   and 5 is a prime number. However there is no evidence to suggest that
   starfish have 5 arms because 5 is a prime number. Indeed, some starfish
   have different numbers of arms. Echinaster luzonicus normally has six
   arms, Luidia senegalensis has nine arms, and Solaster endeca can have
   as many as twenty arms. Why the majority of starfish (and most other
   echinoderms) have five-fold symmetry remains a mystery.

   One example of the use of prime numbers in nature is as an evolutionary
   strategy used by cicadas of the genus Magicicada. These insects spend
   most of their lives as grubs underground. They only pupate and then
   emerge from their burrows after 13 or 17 years, at which point they fly
   about, breed, and then die after a few weeks at most. The logic for
   this is believed to be that the prime number intervals between
   emergences makes it very difficult for predators to evolve that could
   specialise as predators on Magicicadas. If Magicicadas appeared at a
   non-prime number intervals, say every 12 years, then predators
   appearing every 2, 3, 4, 6, or 12 years would be sure to meet them.
   Over a 200-year period, average predator populations during
   hypothetical outbreaks of 14- and 15-year cicadas would be up to 2%
   higher than during outbreaks of 13- and 17-year cicadas. Though small,
   this advantage appears to have been enough to drive natural selection
   in favour of a prime-numbered life-cycle for these insects.

Primes in pop culture

     * In Carl Sagan's novel Contact a message from outer space is sent to
       humanity. This message is hashed into pieces and comes in intervals
       of incremented prime numbers from 2 to 101. The same device was
       later used in an episode of Star Trek: The Next Generation.

   (These fictional alien transmissions might have been inspired by the
   use of prime numbers in constructing the Arecibo message transmitted in
   1974).
     * In Stephen King's novels The Waste Lands and Wizard and Glass, the
       protagonists solve a puzzle using primes in order to ride on the
       sentient and deranged train, Blaine the Mono.

     * In Mark Haddon's novel The Curious Incident of the Dog in the
       Night-time, the main character is fascinated by prime numbers, and
       in the book the chapters are numbered with primes.

     * In the Stargate Atlantis episode "Hot Zone", physicists Rodney
       McKay and Radek Zelenka play the game "Prime or Not Prime" in which
       players randomly name numbers and expect the other player or
       players to determine whether the number is prime.

     * In the film Cube, the protagonists discover that prime numbers
       encoded on passageways represent a possible key to their escape
       from a mysterious facility filled with lethal booby traps.

Trivia

   As a publicity stunt against the Digital Millennium Copyright Act and
   other WIPO Copyright Treaty implementations, some people have tried to
   encode various forms of the DeCSS program in such a way that the
   executable program code becomes a prime number, thus creating illegal
   primes. Such numbers, when converted to binary and executed as a
   computer program, perform acts encumbered by applicable law in one or
   more jurisdictions.

   The largest known prime consisting of a prime number of decimal digits
   is 27,653 · 2^9,167,433 + 1 which has 2,759,677 digits. As of January
   2007 the 6 larger known primes (all Mersenne primes) have numbers of
   digits which are even or divisible by 3.

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