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Encyclopedia > Prime number
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
Superperfect 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
Friendly number
Sociable number
Solitary number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization
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In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The first thirty prime numbers are: A composite number is a positive integer which has a positive divisor other than one or itself. ... A powerful number is a positive integer m that for every prime number p dividing m, p2 also divides m. ... In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. ... An Achilles number is a number that is powerful but not a perfect power. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, an almost perfect number (sometimes also called slightly defective number) is a natural number n such that the sum of all divisors of n (the divisor function σ(n)) is equal to 2n _ 1. ... In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. ... In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. ... In mathematics, a k-hyperperfect number (sometimes just called hyperperfect number) is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i. ... A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. ... In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. ... In mathematics, a primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a natural number that has no semiperfect proper divisor. ... A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ... In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... In mathematics, a highly abundant number is a certain kind of natural number. ... In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. ... In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number. ... A highly composite number is a positive integer which has more divisors than any positive integer below it. ... In mathematics, a superior highly composite number is a certain kind of natural number. ... In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ... The term weird number also refers to a phenomenon in twos complement arithmetic. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... A friendly number is a positive natural number that shares a certain characteristic, to be defined below, with one or more other numbers. ... Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ... In mathematics a solitary number is number which does not have any friends. Two numbers m and n are friends if and only if σ(m)/m = σ(n)/n. ... In mathematics, a sublime number is a positive integer which has a perfect number of positive divisors (including itself), and whose positive divisors add up to another perfect number. ... A harmonic divisor number, or Ore number, is a number whose divisors, averaged in a harmonic mean, results in an integer. ... A frugal number is a natural number that has more digits than the number of digits in its prime factorization (including exponents). ... An equidigital number is a number that has the same number of digits as the number of digits in its prime factorization (including exponents). ... An extravagant number (also known as a wasteful number) is a natural number that has fewer digits than the number of digits in its prime factorization (including exponents). ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... This article is about the concept in number theory. ... ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... This article is about the number one. ... For other uses, see Euclid (disambiguation). ... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC...

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, 113. (sequence A000040 in OEIS)

See the list of prime numbers for a longer list. The number one is by definition not a prime number; see the discussion below under Primality of one. This article does not cite any references or sources. ... This article is about the number. ... Look up five in Wiktionary, the free dictionary. ... Seven Days of Creation - 1765 book, title page 7 (seven) is the natural number following 6 and preceding 8. ... 11 (eleven) is the natural number following 10 and preceding 12. ... 13 (thirteen) is the natural number after 12 and before 14. ... 17 (seventeen) is the natural number following 16 and preceding 18. ... 19 (nineteen) is the natural number following 18 and preceding 20. ... 23 (twenty-three) is the natural number following 22 and preceding 24. ... 29 (twenty-nine) is the natural number following 28 and preceding 30. ... 31 (thirty-one) is the natural number following 30 and preceding 32. ... 37 (thirty-seven) is the natural number following 36 and preceding 38. ... 41 (forty-one) is the natural number following 40 and preceding 42. ... 43 (forty-three) is the natural number following 42 and preceding 44. ... 47 (forty-seven) is the natural number following 46 and preceding 48. ... 53 (fifty-three) is the natural number following 52 and preceding 54. ... 59 (fifty-nine) is the natural number following 58 and preceding 60. ... 61 (sixty-one) is the natural number following 60 and preceding 62. ... 67 (sixty-seven) is the natural number following 66 and preceding 68. ... 71 (seventy-one) is the natural number following 70 and preceding 72. ... 73 (seventy-three) is the natural number following 72 and preceding 74. ... 79 (seventy-nine) is the natural number following 78 and preceding 80. ... 83 (eighty-three) is the natural number following 82 and preceding 84. ... 89 (eighty-nine) is the natural number following 88 and preceding 90. ... 97 is the natural number following 96 and preceding 98. ... 101 (one hundred [and] one) is the natural number following 100 and preceding 102. ... 103 is the natural number following 102 and preceding 104. ... 107 is the natural number following 106 and preceding 108. ... 109 is the natural number following 108 and preceding 110. ... 113 is the natural number following 112 and preceding 114. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... There are infinitely many prime numbers. ...


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


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. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... There is also the Riemann hypothesis for curves over finite fields. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ...


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 bc 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 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 written as the knot sum of two lesser nontrivial knots.

Contents

In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... Not to be confused with Natural number. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... An Eisenstein prime is an Eisenstein integer aω + b that has only two Eisenstein divisors, the complex cube root of unity and aω + b itself. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ... Trefoil knot, the simplest non-trivial knot. ... In knot theory, a prime knot is a knot which is, in a certain sense, indecomposable. ... A trefoil knot. ...

History of prime numbers

The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. It is the predecessor to the modern Sieve of Atkin, which is faster but more complex. The eponymous Sieve of Eratosthenes was created in the 3rd century BC by Eratosthenes, an ancient Greek mathematician.

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 method to compute primes, although the large primes found today with computers are not generated this way. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, the Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. ... Flowcharts are often used to graphically represent algorithms. ... In mathematics, the sieve of Atkin is a fast, modern algorithm for finding all prime numbers up to a specified integer. ... The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period. ... This article is about the Greek scholar of the third century BC. For the ancient Athenian statesman of the fifth century BC, see Eratosthenes (statesman). ... The term ancient Greece refers to the periods of Greek history in Classical Antiquity, lasting ca. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... The pyramids are the most recognizable symbols of the civilization of ancient Egypt. ... An Egyptian fraction is the sum of distinct unit fractions, such as . ... The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts and perhaps our best indication of what ancient Egyptian mathematics might have been like near 2000 BC. They are both written on papyrus. ... The term ancient Greece refers to the periods of Greek history in Classical Antiquity, lasting ca. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, a Mersenne number is a number that is one less than a power of two. ... In mathematics, the Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. ... This article is about the Greek scholar of the third century BC. For the ancient Athenian statesman of the fifth century BC, see Eratosthenes (statesman). ...


After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). A special case of Fermat's theorem may have been known much earlier by the Chinese. Fermat conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4. However, the very next Fermat number 232+1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p - 1, with p a prime. They are called Mersenne primes in his honor. Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Euler redirects here. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... Marin Mersenne, Marin Mersennus or le Père Mersenne (September 8, 1588 – September 1, 1648) was a French theologian, philosopher, mathematician and music theorist. ... In mathematics, a Mersenne number is a number that is one less than a power of two. ...


Euler's work in number theory included many results about primes. He showed the infinite series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … is divergent. In 1747 he showed that the even perfect numbers are precisely the integers of the form 2p-1(2p-1) where the second factor is a Mersenne prime. It is believed no odd perfect numbers exist, but there is still no proof. In the third century BC, Euclid proved the existence of infinitely many prime numbers. ... In mathematics, a series is a sum of a sequence of terms. ...


At the start of the 19th century, Legendre and Gauss independently conjectured that as x tends to infinity, the number of primes up to x is asymptotic to x/log(x), where log(x) is the natural logarithm of x. Ideas of Riemann in his 1859 paper on the zeta-function sketched a program which would lead to a proof of the prime number theorem. This outline was completed by Hadamard and de la Vallée Poussin, who independently proved the prime number theorem in 1896. This page is a candidate for speedy deletion. ... Charles-Jean de la Vallée-Poussin (August 14, 1866 - March 2, 1962) was a Belgian mathematician. ...


Proving a number is prime is not done (for large numbers) by trial division. Many mathematicians have worked on 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 1856),[1] and the generalized Lucas–Lehmer test. More recent algorithms like APRT-CL, ECPP and AKS work on arbitrary numbers but remain much slower. A primality test is an algorithm for determining whether an input number is prime. ... In mathematics, Pépins test is a primality test, which can be used to determine whether a Fermat number is prime. ... Proths theorem states that if p is a prime Proth number ( of the form k * 2^n + 1 with k odd and k < 2^n ), then for some integer a, Where q = ( ( p-1)/2) This means that if you can find some number a, that multiplied it by... In mathematics, the Lucas–Lehmer test is a primality test for Mersenne numbers. ... In computational number theory, the Lucas–Lehmer test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. ... The Adleman-Pomerance-Rumely primality test (APR) is a deterministic algorithm that tests if a positive integer is prime. ... Elliptic Curve Primality Proving is a method based on elliptic curves to prove the primality of a number. ... The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by three Indian scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena on August 6, 2002 in a paper titled PRIMES is in P. The...


For a long time, prime numbers were thought as having no possible application outside of pure mathematics; 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 algorithm. Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... A big random number is used to make a public-key/private-key pair. ... In cryptography, RSA is an algorithm for public-key cryptography. ...


Since 1951 all the largest known primes have been found by computers. The search for ever larger primes has generated interest outside mathematical circles. The Great Internet Mersenne Prime Search and other distributed computing projects to find large primes have become popular in the last ten to fifteen years, while mathematicians continue to struggle with the theory of primes. Graph of number of digits in largest known prime by year - electronic era. ... This article is about the machine. ... The Great Internet Mersenne Prime Search, or GIMPS, is a collaborative project of volunteers, who use Prime95 and MPrime, special software that can be downloaded from the Internet for free, in order to search for Mersenne prime numbers. ... Distributed computing is a method of computer processing in which different parts of a program are run simultaneously on two or more computers that are communicating with each other over a network. ...


Primality of one

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 labeling 1 a prime, such as the work of Stern and Zeisel. Derrick Norman Lehmer's list of primes up to 10006721, reprinted as late as 1956,[2] started with 1 as its first prime.[3] Henri Lebesgue is said to be the last professional mathematician to call 1 prime. The change in label occurred so that the fundamental theorem of arithmetic, as stated, is valid, i.e., “each number has a unique factorization into primes”[4][5][6] Moritz Abraham Stern (1807-1894) was a German mathematician. ... Derrick Norman Lehmer (27 July 1867, Somerset, Indiana, USA — 8 September 1938 in Berkeley, California, USA) was an American mathematician and number theorist. ... Henri Lebesgue Henri Léon Lebesgue (June 28, 1875, Beauvais – July 26, 1941, Paris) was a French mathematician, most famous for his theory of integration. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ...


Prime divisors

Illustration showing that 11 is a prime number while 12 is not.

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 way which is unique except possibly for the order of the prime factors. The same prime factor may occur multiple times. Primes can thus be considered the “basic building blocks” of the natural numbers. For example, we can write Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

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. Prime decomposition redirects here. ...


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

  • When written in base 10, all prime numbers except 2 and 5 end in 1, 3, 7 or 9. (Numbers ending in 0, 2, 4, 6 or 8 represent multiples of 2 and numbers ending in 0 or 5 represent multiples of 5.)
  • 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.
  • If p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is 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 (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 px, then S(x) = ln ln x + O(1) for x → ∞.
  • If G is a finite group and pn is the highest power of the prime p which divides the order of G, then G has a subgroup of order pn. (Sylow theorems.)
  • If G is a finite group and p is a prime number dividing the order of G, then G contains an element of order p. (Cauchy Theorem)
  • 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 in base ten, 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:
Evaluating this identity at different integers provides an infinite number of products over the primes whose values can be calculated, the first two being
  • If p > 1, the polynomial is irreducible over Z/pZ if and only if p is prime.
  • All prime numbers above 3 are of form 6n − 1 or 6n + 1, because all other numbers are divisible by 2 or 3. Generalizing this, all prime numbers above q are of form q#·n + m, where 0 < m < q, and m has no prime factor ≤ q.

Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to... Euclids lemma is a generalisation of Proposition 30 of Book VII of Euclids Elements. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... ↔ ⇔ ≡ logical symbols representing iff. ... The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... For factorial rings in mathematics, see unique factorisation domain. ... In mathematics, Wilsons theorem (also known as Al-Haythams theorem) states that p > 1 is a prime number if and only if (see factorial and modular arithmetic for the notation). ... Bertrands postulate states that if n > 3 is an integer, then there always exists at least one prime number p with n < p < 2n âˆ’ 2. ... In mathematics, a series is a sum of a sequence of terms. ... In the third century BC, Euclid proved the existence of infinitely many prime numbers. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ... In number theory, Dirichlets theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n > 0, or in other words: there are infinitely many primes which are congruent to a modulo d. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... This picture illustrates how the hours on a clock form a group under modular addition. ... The Sylow theorems of group theory, named after Ludwig Sylow, form a partial converse to Lagranges theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the... -1... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... For other uses, see Decimal (disambiguation). ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... For n ≥ 2, the primorial n# is the product of all prime numbers less than or equal to n. ...

Classification of prime numbers

Two ways of classifying prime numbers, class n+ and class n−, were studied by Paul Erdős and John Selfridge. Paul ErdÅ‘s (Hungarian: ErdÅ‘s Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... John Selfridge is a mathematician who has contributed to the field of analytic number theory. ...


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: Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...

Consider any finite set of primes. Multiply all of them together and add one (see Euclid number). The resulting number is not divisible by any of the primes in the finite set we considered, because dividing by any of these would give a remainder of one. Because all non-prime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. Either way, there is at least one more prime that was not in the finite set we started with. This argument applies no matter what finite set we began with. So there are more primes than any given finite number. To meet Wikipedias quality standards, this article or section may require cleanup. ...

This previous argument explains why the product P of finitely many primes plus 1 must be divisible by some prime not among those finitely many primes (possibly itself).


The proof is sometimes phrased in a way that leads the student to conclude that P + 1 must itself be prime, and think that Euclid's proof says the prime product plus 1 is always prime. The simple example of (2 · 3 · 5 · 7 · 11 · 13) + 1 = 30,031 = 59 · 509 (both primes) shows that this is not the case. In fact, any set of primes which does not include 2 will have an odd product. Adding 1 to this product will always produce an even number, which will be divisible by 2 (and therefore not be prime). See also Euclid's theorem. Euclids Theorem is generally a reference to the theorem (often credited to Euclid) which demonstrates the existence of an infinite number of prime numbers. ...


Other mathematicians have given other proofs. One of these (due to Euler) shows that the sum of the reciprocals of all prime numbers diverges. Another proof based on Fermat numbers was given by Goldbach.[7] Kummer's is particularly elegant[8] and Harry Furstenberg provides one using general topology.[9][10] Euler redirects here. ... In the third century BC, Euclid proved the existence of infinitely many prime numbers. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... Christian Goldbach (March 18, 1690 - November 20, 1764), was a Prussian mathematician, who was born in Königsberg, Prussia, as son of a pastor. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Hillel (Harry) Furstenberg is an Israeli mathematician. ... In mathematics, Hillel Furstenbergs proof of the infinitude of primes is a celebrated topological proof that the integers contain infinitely many prime numbers. ...


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 possible to compute each prime up to x. Values as large as π(1020) can be calculated quickly and accurately with modern computers. Thus, e.g., π(100000) = 9592, and π(1020) = 2,220,819,602,560,918,840. In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. ... Flowcharts are often used to graphically represent algorithms. ...


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. In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...


Location of prime numbers

Finding prime numbers

The ancient Sieve of Eratosthenes is a simple way to compute all prime numbers up to a given limit, by making a list of all integers and repeatedly striking out multiples of already found primes. The modern Sieve of Atkin is more complicated, but faster when properly optimized. In mathematics, the Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. ... In mathematics, the sieve of Atkin is a fast, modern algorithm for finding all prime numbers up to a specified integer. ...


In practice one often 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. Probability is the likelihood that something is the case or will happen. ... A primality test is an algorithm for determining whether an input number is prime. ... A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime. ...


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. In mathematics, a quotient is the end result of a division problem. ...


Primality tests

Main article: primality test

A primality test algorithm is an algorithm which tests a number for primality, i.e. whether the number is a prime number. A primality test is an algorithm for determining whether an input number is prime. ... A primality test is an algorithm for determining whether an input number is prime. ...

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. The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by three Indian scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena on August 6, 2002 in a paper titled PRIMES is in P. The... The Fermat primality test is a probabilistic test to determine if a number is composite or probably prime. ... In mathematics, the Lucas-Lehmer test is a primality test for Mersenne numbers. ... The Solovay-Strassen primality test, developed by Robert M. Solovay and Volker Strassen, is a probabilistic test to determine if a number is composite or probably prime. ... The Miller-Rabin primality test or Rabin-Miller primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay-Strassen primality test. ... Elliptic Curve Primality Proving is a method based on elliptic curves to prove the primality of a number. ... In number theory, a probable prime (PRP) is an integer that satisfies a condition also satisfied by all prime numbers. ... In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence bn âˆ’ 1 ≡ 1 (mod n) for all integers b which are relatively prime to n (see modular arithmetic). ... A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime. ...


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). The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by three Indian scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena on August 6, 2002 in a paper titled PRIMES is in P. The... In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ...


Formulas yielding prime numbers

Main article: formula for primes

There is no known formula for primes which is more efficient at finding primes than the methods mentioned above under “Finding prime numbers”. In mathematics, a formula for primes is a formula generating the prime numbers, exactly and without exception. ... In mathematics, a formula for primes is a formula generating the prime numbers, exactly and without exception. ...


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. In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ...


There is no polynomial, even in several variables, that takes only prime values. For example, the curious polynomial in one variable f(n) = n2n + 41 yields primes for n = 0,…, 40,43 but f(41) and f(42) are composite. However, there are polynomials in several variables, whose positive values as the variables take all positive integer values are exactly the primes. In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...


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 formulas 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: In mathematics, primorial primes are prime numbers of the form pn# Â± 1, where: pn# is the primorial of pn. ... For n ≥ 2, the primorial n# is the product of all prime numbers less than or equal to n. ... A factorial prime is a number that is one less or one more than a factorial and is also a prime number. ... For factorial rings in mathematics, see unique factorisation domain. ...

n! − 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, … (sequence A002982 in OEIS)
n! + 1 is prime for n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, … (sequence A002981 in OEIS)

The largest known primorial prime is Π(392113) + 1, found by Heuer in 2001.[11] The largest known factorial prime is 34790! − 1, found by Marchal, Carmody and Kuosa in 2002.[12] It is not known whether there are infinitely many primorial or factorial primes. The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...


Primes of the form 2p − 1, where p is a prime number, are known as Mersenne primes, while primes of the form 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: In mathematics, a Mersenne number is a number that is one less than a power of two. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... A prime number p is called a Sophie Germain prime if 2p + 1 is also prime. ...

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. In mathematics, a Wieferich prime is prime number p such that p² divides 2p − 1 − 1; compare this with Fermats little theorem, which states that every prime p divides 2p − 1 − 1. ... In mathematics, a Wilson prime is a certain kind of prime number. ... In mathematics, a Wall-Sun-Sun prime is a certain kind of prime number. ... In mathematics, a Wolstenholme prime is a certain kind of prime number. ... In mathematics, a unique prime is a certain kind of prime number. ... This can be abbreviated to NSW, which is also the abbreviation of the state of New South Wales in Australia. ... In mathematics, Smarandache-Wellin numbers are special integers; the n-th Smarandache-Wellin number is defined as the concatenation of the first n prime numbers written in decimal notation. ... In mathematics, a prime number of the form (2p + 1) / 3 for a prime number p is called a Wagstaff prime; they are related to the New Mersenne conjecture. ... In mathematics, a supersingular prime is a certain kind of prime number. ... For the movie, see Palindromes (film). ... A palindromic prime is a prime number that is also a palindromic number. ... 357686312646216567629137 is the largest prime number that is left-truncatable in decimal, meaning that the number, as well as all the numbers obtained by successively removing the first digit at the left of the number are prime: 357686312646216567629137, 57686312646216567629137, 7686312646216567629137, ..., 9137, 137, 37 and 7 are all prime. ...

There are infinitely many prime numbers. ...

The distribution of the prime numbers

Further information: Prime number theorem
The distribution of all the prime numbers in the range of 1 to 76,800, from left to right and top to bottom, where each pixel represents a number. Black pixels mean that number is prime and white means it is 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 behavior[clarify]. Leonhard Euler commented In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... Image File history File links No higher resolution available. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Euler redirects here. ...

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 Paul ErdÅ‘s (Hungarian: ErdÅ‘s Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ...

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 “Einstein” redirects here. ... Don Bernhard Zagier (1951 - ) is an American mathematician. ...

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 behavior, and that they obey these laws with almost military precision.

(Havil 2003, p. 171)


Additional image with 2310 columns is linked here, preserving multiples of 2,3,5,7,11 in respective columns. Image File history File links Size of this preview: 800 × 196 pixelsFull resolution (2310 × 565 pixel, file size: 58 KB, MIME type: image/png) Graphical depiction of the prime numbers in the range 1 to 1299827, where each pixel represents an integer, and the white pixels represent primes. ...


Gaps between primes

Main article: Prime gap

Let pn denote the nth prime number (i.e. p1 = 2, p2 = 3, etc.). The gap gn between the consecutive primes pn and pn + 1 is the difference between them, i.e. The n-th prime gap (short for prime number gap), denoted gn, is the difference between the n+1-th and n-th prime number, pn. ...

gn = pn + 1pn.

We have g1 = 3 − 2 = 1, g2 = 5 − 3 = 2, g3 = 7 − 5 = 2, g4 = 11 − 7 = 4, and so on. The sequence (gn) of prime gaps has been extensively studied.


For any natural number N larger than 1, the sequence (for the notation N! read factorial) For factorial rings in mathematics, see unique factorisation domain. ...

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 gn > N. (Choose n so that pn 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 gn/pn approaches zero as n approaches infinity. Note also that the twin prime conjecture asserts that gn = 2 for infinitely many integers n. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ...


Location of the largest known prime

The largest known prime, as of August 2007, is 232,582,657 − 1 (this number is 9,808,358 digits long); it is the 44th known Mersenne prime. M32582657 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 software on a peak of 700 university computers for 9 years. Graph of number of digits in largest known prime by year - electronic era. ... In mathematics, a Mersenne number is a number that is one less than a power of two. ... Image File history File links WikiNews-Logo. ... Wikinews is a free-content news source and a project of the Wikimedia Foundation. ... Image File history File links WikiNews-Logo. ... Wikinews is a free-content news source and a project of the Wikimedia Foundation. ... For other uses, see Number (disambiguation). ... In mathematics, a Mersenne number is a number that is one less than a power of two. ... is the 247th day of the year (248th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... The meaning of the word professor (Latin: [1]) varies. ... The University of Central Missouri (formerly Central Missouri State University) is a four-year public institution in Warrensburg, Missouri a town of 16,3340 in Johnson County, Missouri. ... The Great Internet Mersenne Prime Search, or GIMPS, is a collaborative project of volunteers, who use Prime95 and MPrime, special software that can be downloaded from the Internet for free, in order to search for Mersenne prime numbers. ...


The next two largest known primes are also Mersenne Primes: M30,402,457 = 230,402,457 − 1 (43rd known Mersenne prime, 9,152,052 digits long) and M25,964,951 = 225,964,951 − 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. In mathematics, the Lucas–Lehmer test is a primality test for Mersenne numbers. ...


The largest known prime that is not a Mersenne prime is 19,249 × 213,018,586 + 1 (3,918,990 digits), a Proth number. This is also the seventh largest known prime of any form. It was found on March 26, 2007 by the Seventeen or Bust project and it brings them one step closer to solving the Sierpinski problem. Proths theorem states that if p is a prime Proth number ( of the form k * 2^n + 1 with k odd and k < 2^n ), then for some integer a, Where q = ( ( p-1)/2) This means that if you can find some number a, that multiplied it by... March 26 is the 85th day of the year (86th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) was a common year starting on Monday of the Gregorian calendar in the 21st century. ... Seventeen or Bust is a distributed computing project to solve the last seventeen cases in the Sierpinski problem. ... In number theory, a Sierpinski number is an odd natural number k such that integers of the form k2n + 1 are composite (i. ...


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 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].


Awards for finding primes

The Electronic Frontier Foundation (EFF) has offered a US$100,000 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. EFF Logo The Electronic Frontier Foundation (EFF) is an international non-profit advocacy and legal organization based in the United States with the stated purpose of being dedicated to preserving free speech rights such as those protected by the First Amendment to the United States Constitution in the context of...


The RSA Factoring Challenge offered prizes up to US$200,000 for finding the prime factors of certain semiprimes of up to 2048 bits. However, the challenge was closed in 2007 after much smaller prizes for smaller semiprimes had been paid out.[13] The RSA Factoring Challenge is a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. ... In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ...


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, …}. In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... UFD redirects here, but this abbreviation can also mean USB flash drive, an electronic device. ...


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. A Gaussian integer is a complex number whose real and imaginary part are both integers. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ...


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), … In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...


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. In mathematics, class field theory is a major branch of algebraic number theory. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Look up valuation in Wiktionary, the free dictionary. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ...


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. Trefoil knot, the simplest non-trivial knot. ... A trefoil knot. ... Trefoil knot, the simplest non-trivial knot. ...


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. There is also the Riemann hypothesis for curves over finite fields. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...


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.
  • 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 conjectured there are infinitely many primes of the form n2 + 1.[15]
  • Cramér's conjecture: . 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 consecutive primes greater than 2.

All four of Landau's problems from 1912 are listed above and still unsolved: Goldbach, twin primes, Legendre, n2+1 primes. To meet Wikipedias quality standards, this article or section may require cleanup. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... In number theory, Goldbachs weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that: Every odd number greater than 7 can be expressed as the sum of three odd primes. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ... A twin prime is a prime number that differs from another prime number by two. ... Alphonse de Polignac (1817 – 1890) was a French mathematician who made the conjecture that for every natural number k, there are infinitely many prime pairs which have a distance of 2k. ... In mathematics, any integer (whole number) is either even or odd. ... In mathematics, a Mersenne number is a number that is one less than a power of two. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In mathematics, Schinzels hypothesis H is a very broad generalisation of conjectures such as the twin prime conjecture. ... A Fibonacci prime is a Fibonacci number that is prime. ... Adrien-Marie Legendre conjectured that there is a prime number between n² and (n+1)² for every integer n > 0. ... In mathematics, Cramérs conjecture, formulated by the Swedish mathematician Harald Cramér in 1937, states that where pn denotes the nth prime number and log is the natural logarithm. ... In mathematics, Brocards conjecture is a conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, for n > 1, where pn is the nth prime number. ... At the 1912 International Congress of Mathematicians in Cambridge, Edmund Landau listed four basic problems about primes. ...


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.[17] 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. Godfrey Harold Hardy FRS (February 7, 1877 Cranleigh, Surrey, England [1] – December 1, 1947 Cambridge, Cambridgeshire, England [2]) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... Public key cryptography is a form of cryptography which generally allows users to communicate securely without having prior access to a shared secret key, by using a pair of cryptographic keys, designated as public key and private key, which are related mathematically. ... In computer science, a hash table is a data structure that speeds up searching for information by a particular aspect of that information, called a key. ... A pseudorandom number generator (PRNG) is an algorithm to generate a sequence of numbers that approximate the properties of random numbers. ...


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. A series of three rotors from an Enigma machine, used by Germany during World War II In cryptography, a rotor machine is an electro-mechanical device used for encrypting and decrypting secret messages. ... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ... Full Cycle Recordings is a record label based in Bristol, England specialising in drum and bass. ...


Public-key cryptography

Several public-key cryptography algorithms, such as RSA, are based on large prime numbers (for example with 512 bits). Public key cryptography is a form of cryptography which generally allows users to communicate securely without having prior access to a shared secret key, by using a pair of cryptographic keys, designated as public key and private key, which are related mathematically. ... In cryptography, RSA is an algorithm for public-key cryptography. ... This article is about the unit of information. ...


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. Orders Brisingida (100 species[1]) Forcipulatida (300 species[2]) Paxillosida (255 species[3]) Notomyotida (75 species[4]) Spinulosida (120 species[5]) Valvatida (695 species[6]) Velatida (200 species[7]) For other uses, see Starfish (disambiguation). ... Classes Asteroidea Concentricycloidea Crinoidea Echinoidea Holothuroidea Ophiuroidea Echinoderms (Echinodermata) is a phylum of marine animals found in the ocean at all depths. ... The elaborate patterns on the wings of butterflies are one example of bilateral symmetry. ...


One example of the use of prime numbers in nature is as an evolutionary strategy used by cicadas of the genus Magicicada.[18] 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.[19] 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.[20] Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects. For other uses, see Cicada (disambiguation). ... Species See text. ... A larval insect A larva (Latin; plural larvae) is a juvenile form of animal with indirect development, undergoing metamorphosis (for example, insects or amphibians). ...


There is speculation that the zeros of the zeta function are connected to the energy levels of complex quantum systems. [21]


Prime numbers in the arts and literature

Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as La Nativité du Seigneur (1935) and Quatre études de rythme (1949-50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in one of the études. According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations". [22] A composer is a person who writes music. ... Olivier Messiaen It has been suggested that List of students of Olivier Messiaen be merged into this article or section. ...


In his science fiction novel Contact, later made into a film of the same name, the NASA scientist Carl Sagan suggested that prime numbers could be used as a means of communicating with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975. [23] Contact is a science fiction novel written by Carl Sagan and published in 1985. ... Contact is a 1997 science fiction film adapted from the novel by Carl Sagan. ... For other uses, see NASA (disambiguation). ... Insert non-formatted text here Carl Edward Sagan (November 9, 1934 – December 20, 1996) was an American astronomer and astrobiologist and a highly successful popularizer of astronomy, astrophysics, and other natural sciences. ... Professor Frank Drake Frank Drake (born May 28, 1930, Chicago, Illinois) is an American astronomer and astrophysicist. ...


Tom Stoppard's award-winning 1993 play Arcadia was a conscious attempt to discuss mathematical ideas on the stage. In the opening scene, the 13 year old heroine puzzles over Fermat's last theorem, a theorem involving prime numbers. [24] [25] [26] Sir Tom Stoppard, OM, CBE (born as Tomáš Straussler on July 3, 1937)[1] is an Academy Award winning British playwright of more than 24 plays. ... Arcadia is a play by Tom Stoppard which first opened at the Royal National Theatre in London on 13 April 1993 and has played at many theatres since. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ...


Many films reflect a popular fascination with the mysteries of prime numbers and cryptography: films such as The Cube, Sneakers, The Mirror Has Two Faces and A Beautiful Mind, based on the biography of the mathematician and Nobel laureate John Forbes Nash by Sylvia Nasar.[27] [28] The Cube is a Microplex Cinema and venue in central Bristol. ... This article or section contains a plot summary that is overly long or excessively detailed compared to the rest of the article. ... The Mirror Has Two Faces (1996) is a romantic comedy movie starred and directed by Barbra Streisand. ... A Beautiful Mind is a 2001 American biographical film about John Forbes Nash, the Nobel Laureate (Economics) mathematician. ... John Forbes Nash, Jr. ... Sylvia Nasar (born 1947 in Bavaria) is an American journalist and writer. ...


See also

Full Cycle Recordings is a record label based in Bristol, England specialising in drum and bass. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... Prime decomposition redirects here. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. ... A primality test is an algorithm for determining whether an input number is prime. ... A prime power is a positive integer power of a prime. ... In mathematical physics, the primon gas or free Riemann gas is a toy model illustrating in a simple way some correspondences between number theory and ideas in quantum field theory and dynamical systems. ... For n ≥ 2, the primorial n# is the product of all prime numbers less than or equal to n. ... A sphenic number is a positive integer that is the product of three distinct prime factors. ... The Ulam spiral, or prime spiral (in other languages also called the Ulam cloth) is a simple method of graphing the prime numbers that reveals a pattern which has never been fully explained. ... There are infinitely many prime numbers. ...

Notes

  1. ^ The Largest Known Prime by Year: A Brief History Prime Curios!: 17014…05727 (39-digits)
  2. ^ Hans Riesel, Prime Numbers and Computer Methods for Factorization. New York: Springer (1994): 36
  3. ^ Richard K. Guy & John Horton Conway, The Book of Numbers. New York: Springer (1996): 129 - 130
  4. ^ Gowers, T (2002). Mathematics: A Very Short Introduction. Oxford University Press, 118. ISBN 0-19-285361-9. “The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes” 
  5. ^ ""Why is the number one not prime?"". Retrieved 2007-10-02.
  6. ^ ""Arguments for and against the primality of 1".
  7. ^ Letter in Latin from Goldbach to Euler, July 1730.
  8. ^ P. Ribenboim: The Little Book of Bigger Primes, second edition, Springer, 2004, p. 4.
  9. ^ Furstenberg, Harry. (1955). "On the infinitude of primes". Amer. Math. Monthly 62 (5): 353. doi:10.2307/2307043. 
  10. ^ Furstenberg's proof that there are infinitely many prime numbers. Everything2. Retrieved on 2006-11-26.
  11. ^ The Top Twenty: Primorial
  12. ^ The Top Twenty: Factorial
  13. ^ The RSA Factoring Challenge — RSA Laboratories
  14. ^ E.g., see Guy, Richard K. (1981), Unsolved Problems in Number Theory, Springer-Verlag , problem A3, pp. 7–8.
  15. ^ Eric W. Weisstein, Landau's Problems at MathWorld.
  16. ^ Caldwell, Chris, The Top Twenty: Lucas Number at The Prime Pages.
  17. ^ Hardy, G.H. (1940). A Mathematician's Apology. Cambridge University Press. ISBN 0-521-42706-1. “No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years” 
  18. ^ Goles, E., Schulz, O. and M. Markus (2001). "Prime number selection of cycles in a predator-prey model", Complexity 6(4): 33-38
  19. ^ Paulo R. A. Campos, Viviane M. de Oliveira, Ronaldo Giro, and Douglas S. Galvão. (2004). "Emergence of Prime Numbers as the Result of Evolutionary Strategy". Phys. Rev. Lett. 93. doi:10.1103/PhysRevLett.93.098107. Retrieved on 2006-11-26. 
  20. ^ Invasion of the Brood. The Economist (May 6, 2004). Retrieved on 2006-11-26.
  21. ^ Ivars Peterson (June 28, 1999). The Return of Zeta. MAA Online. Retrieved on 2008-3-14.
  22. ^ The Messiaen companion', ed. Peter Hill, Amadeus Press, 1994. ISBN 0-931340-95-0
  23. ^ Carl Pomerance, Prime Numbers and the Search for Extraterrestrial Intelligence, Retrieved on December 22, 2007
  24. ^ Tom Stoppard, Arcadia, Faber and Faber, 1993. ISBN 0-571-16934-1.
  25. ^ The Cambridge Companion to Tom Stoppard, ed. Katherine E. Kelly, Cambridge University Press, 2001. ISBN 0521645921
  26. ^ The Mathematics of Arcadia, an event involving Tom Stoppard and MSRI in the University of California, Berkeley
  27. ^ Music of the Spheres, Marcus du Sautoy's selection of films featuring prime numbers
  28. ^ A Beautiful Mind

William Timothy Gowers (born November 20, 1963, Wiltshire, United Kingdom) is a British mathematician. ... Oxford University Press (OUP) is a highly-respected publishing house and a department of the University of Oxford in England. ... For other uses, see Latins and Latin (disambiguation). ... The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Everything2, Everything2, or E2 for short, is a collaborative Web-based community consisting of a database of interlinked user-submitted written material. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 330th day of the year (331st in leap years) in the Gregorian calendar. ... Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... The Prime Pages is a website about prime numbers maintained by Prof. ... Godfrey Harold Hardy FRS (February 7, 1877 Cranleigh, Surrey, England [1] – December 1, 1947 Cambridge, Cambridgeshire, England [2]) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... A Mathematicians Apology is a 1940 essay by British mathematician G. H. Hardy. ... The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. ... Physical Review Letters is one of the most prestigious journals in physics. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 330th day of the year (331st in leap years) in the Gregorian calendar. ... The Economist is an English-language weekly news and international affairs publication owned by The Economist Newspaper Ltd and edited in London. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 330th day of the year (331st in leap years) in the Gregorian calendar. ... is the 179th day of the year (180th in leap years) in the Gregorian calendar. ... Events of 2008: (EMILY) Me Lesley and MIley are going to China! This article is about the year. ... One of the top number theorists of our time, Carl Pomerance received his PhD from Harvard University in 1972 and immediately joined the faculty at the University of Georgia, becoming full professor in 1982. ... is the 356th day of the year (357th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) was a common year starting on Monday of the Gregorian calendar in the 21st century. ... The Mathematical Sciences Research Institute (MSRI), founded in 1982, is a mathematical research institution whose funding sources include the National Science Foundation. ... Sather Tower (the Campanile) looking out over the San Francisco Bay and Mount Tamalpais. ... Marcus du Sautoy (1965–) is a Professor of Mathematics at the University of Oxford. ...

References

  • John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press; 448 pages
  • Wladyslaw Narkiewicz, The development of prime number theory. From Euclid to Hardy and Littlewood. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000.
  • H. Riesel, Prime Numbers and Computer Methods for Factorization, 2nd ed., Birkhäuser 1994.
  • Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. HarperCollins; 352 pages. ISBN 0-06-621070-4. The Music of Primes website.
  • Karl Sabbagh, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics. Farrar, Straus and Giroux; 340 pages

External links

The Prime Pages is a website about prime numbers maintained by Prof. ...

Prime number generators & calculators



  Results from FactBites:
 
Prime number - Wikipedia, the free encyclopedia (4339 words)
The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers.
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.
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.
Prime number - definition of Prime number in Encyclopedia (2598 words)
Prime numbers are of fundamental importance in number theory.
Numbers all of whose prime factors are the same are prime powers.
In number theory itself, one talks of "probable primes", integers which, by virtue of having passed a certain test, are considered to be probably prime.
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