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.

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 2^p - 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 2^p-1(2^p-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.

• 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, 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 p ≤ x, then S(x) = ln ln x + O(1) for x → ∞.

• In every arithmetic progression a, a + q, a + 2q, a + 3q,… where the positive integers a and q 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 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. ...

• AKS primality test
• Fermat primality test
• Lucas-Lehmer test
• Solovay-Strassen primality test
• Miller-Rabin primality test
• Elliptic curve primality proving

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) = n² − n + 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 2^p − 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. ...

• For a list of special classes of prime numbers‎ see List of prime numbers‎

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 p_n denote the nth prime number (i.e. p₁ = 2, p₂ = 3, etc.). The gap g_n between the consecutive primes p_n and p_{n + 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. ...

g_n = p_{n + 1} − p_n.

We have g₁ = 3 − 2 = 1, g₂ = 5 − 3 = 2, g₃ = 7 − 5 = 2, g₄ = 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