query: what is the lowest prime number?

title: Prime number

text: leads to a different but equivalent definition of the primes: they are the numbers with exactly two positive divisors, 1 and the number itself. Yet another way to say the same thing is that a number formula_1 is prime if it is greater than one and if none of the numbers formula_10 divides formula_1 evenly. The first 25 prime numbers (all the prime numbers less than 100) are: No even number formula_1 greater than 2 is prime because any such number can be expressed as the product formula_13. Therefore, every prime number other than 2 is an odd number, and

title: Prime number

text: in a generalized way like prime numbers include prime elements and prime ideals. A natural number (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it. The numbers greater than 1 that are not prime are called composite numbers. In other words, formula_1 is prime if formula_1 items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange formula_1 dots into

title: Prime number

text: of a number can be replaced with that of an ideal, a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements. "Prime ideals", which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, 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), … The fundamental theorem of arithmetic generalizes

title: Prime number

text: of such a domain is the Gaussian integers formula_179, the ring of complex numbers of the form formula_180 where formula_181 denotes the imaginary unit and formula_34 and formula_35 are arbitrary integers. Its prime elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes formula_184 and formula_185. Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not. This

title: 260 (number)

text: AD 260 and 260 BC. 261 = 3·29, lucky number, nonagonal number, Harshad number, unique period in base 2, number of possible unfolded tesseract patterns. 261 was once the lowest number not to have its own Wikipedia page, this making it a candidate for the lowest Uninteresting Number according to the definition given by Alex Bellos. As of 27 March 2018, the lowest natural number without its own Wikipedia page was 262. 262 = 2·131, meandric number, open meandric number, untouchable number, happy number, palindrome number, semiprime 263 is a prime, safe prime, happy number, sum of five consecutive primes

title: Prime number

text: the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number. By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number. In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be

title: Prime number

text: a rectangular grid that is more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. and are both composite. The divisors of a natural number formula_1 are the numbers that divide formula_1 evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This idea

title: Largest known prime number

text: Largest known prime number The largest known prime number () is 2 − 1, a number with 23,249,425 digits. It was found by the Great Internet Mersenne Prime Search (GIMPS) in 2017. Euclid proved that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers. Many of the largest known primes are Mersenne primes. , the seven largest known primes are Mersenne primes. The last sixteen record primes were Mersenne primes. The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests

title: Prime number

text: are distinct Fermat primes. Likewise, a regular formula_1-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of formula_1 are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form formula_241. It is possible to partition any convex polygon into formula_1 smaller convex polygons of equal area and equal perimeter, when formula_1 is a power of a prime number, but this is not known for other values of formula_1. Beginning with the work of Hugh Montgomery and Freeman Dyson in

title: Prime number theorem

text: function that gives the number of primes less than or equal to , for any real number . For example, because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that is a good approximation to , in the sense that the limit of the "quotient" of the two functions and as increases without bound is 1: known as the asymptotic law of distribution of prime numbers. Using asymptotic notation this result can be restated as This notation (and the theorem) does "not" say anything about the

title: Prime number

text: in several variables, that takes "only" prime values. However, there are numerous expressions that do encode all primes, or only primes. One possible formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. There is also a set of Diophantine equations in nine 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. Other

title: Largest known prime number

text: the progression of the largest known prime number in ascending order. Here M = 2 − 1 is the Mersenne number with exponent "n". The longest record-holder known was M = 524,287, which was the largest known prime for 144 years. No records are known before 1456. GIMPS found the fourteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world. A list of the 5,000 largest known primes is maintained by Chris K. Caldwell, of which the twenty largest are listed below. Largest known prime number The largest known prime number () is

title: Prime number

text: Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave

title: Prime number

text: instance the Solovay–Strassen primality test on a given number formula_20 chooses a number formula_34 randomly from formula_42 to formula_210 and uses modular exponentiation to check whether formula_211 is divisible by formula_20. If so, it answers yes and otherwise it answers no. If formula_20 really is prime, it will always answer yes, but if formula_20 is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2. If this test is repeated formula_1 times on the same number, the probability that a composite number could pass the test every time is at most formula_216. Because

title: Prime number

text: a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number formula_1, called trial division, tests whether formula_1 is a multiple of any integer between 2 and formula_3. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available

title: Prime number

text: of arithmetic, and shows the close connection between the zeta function and the prime numbers. It leads to another proof that there are infinitely many primes: if there were only finitely many, then the sum-product equality would also be valid at formula_113, but the sum would diverge (it is the harmonic series formula_114) while the product would be finite, a contradiction. The Riemann hypothesis states that the zeros of the zeta-function are all either negative even numbers, or complex numbers with real part equal to 1/2. The original proof of the prime number theorem was based on a weak form

title: Prime number

text: 2 to , which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime. Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers. However, trial division is still used, with a smaller limit than the square root on the divisor size, to quickly discover composite numbers with small factors, before using more complicated methods on the numbers that pass this filter. Before computers, mathematical tables

title: 64 (number)

text: 64 (number) 64 (sixty-four) is the natural number following 63 and preceding 65. Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers. It is also a dodecagonal number and a centered triangular number. 64 is also the first whole number that is both a perfect square and a perfect cube. Since

title: 31 (number)

text: 31 is a centered triangular number, the lowest prime centered pentagonal number and a centered decagonal number. For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals. At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31, making 31 a self number. 31 is a repdigit in base 5 (111), and base 2 (11111). The numbers 31, 331, 3331, , , , and are all prime. For a time it

title: Sierpinski number

text: "what is the smallest prime Sierpiński number", and there is an ongoing "Prime Sierpiński search" which tries to prove that "271129" is the first Sierpiński number which is also a prime. The prime values of "k" less than "271129" for which a prime of the form "k"2+1 is not known are: The first two, being less than 78557, are also unsolved cases of the (non-prime) Sierpiński problem described above. Suppose that both preceding Sierpiński problems had finally been solved, showing that 78557 is the smallest Sierpiński number and that 271129 is the smallest prime Sierpiński number. This still leaves unsolved

title: Prime number

text: story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with Asperger syndrome. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel "The Solitude of Prime Numbers", in which they are portrayed as "outsiders" among integers. Prime number A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because

title: Prime number

text: Prime number A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 6 is composite because it is the product of two numbers () that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either

title: Prime number

text: a product of prime numbers is called a "prime factorization" of the number. For example: The terms in the product are called prime factors. The same prime factor may occur more than once; this example has two copies of the prime factor formula_28. When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for instance, in the second way of writing the product above, formula_29 denotes the square or second power of formula_28. The central importance of prime numbers to number theory and mathematics in general stems from the "fundamental

title: Prime number

text: the points has large area. Another example is Eisenstein's criterion, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square. The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a finite field with a prime number of elements, whence

title: Prime number

text: such as the Meissel–Lehmer algorithm can compute exact values of formula_88 faster than it would be possible to list each prime up to formula_1. The prime number theorem states that formula_88 is asymptotic to formula_94, which is denoted as and means that the ratio of formula_88 to the right-hand fraction approaches 1 as formula_1 grows to infinity. This implies that the likelihood that a randomly chosen number less than formula_1 is prime is (approximately) inversely proportional to the number of digits in formula_1. It also implies that the formula_1th prime number is proportional to formula_101 and therefore that the average

title: Prime number

text: for numbers of special forms, such as Mersenne numbers. the largest known prime number has 23,249,425 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.

title: Prime number

text: a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. Another type of problem concerns prime gaps, the differences between consecutive primes. The existence of arbitrarily large prime gaps can be seen by noting that the sequence formula_60 consists of formula_61 composite numbers, for any natural number formula_1. However, large prime gaps occur much earlier than this argument shows. For example, the first prime gap of length 8 is between the primes 89 and

title: This Is What the Truth Feels Like

text: outside the top 10. The following week, "This Is What the Truth Feels Like" dipped 54 positions, the week's largest drop. In Australia, the album debuted at number six, becoming her third consecutive top-ten solo album. "This Is What the Truth Feels Like" was the week's second highest debut, and was present for two weeks before departing the chart. On France's SNEP albums chart, it entered at number 44, becoming her lowest debut behind "The Sweet Escape"s peak at number 33. The album fared moderately well on Switzerland's Hitparade, where it became her second highest performing album, peaking at number

title: Weakly prime number

text: Weakly prime number In number theory, a prime number is called weakly prime if it becomes not prime when any one of its digits is changed to every single other digit. Decimal digits are usually assumed. The first weakly prime numbers are: For the first of these, each of the 54 numbers 094001, 194001, 394001, ..., 294009 are composite. A weakly prime base-"b" number with "n" digits must produce ("b"−1) × "n" composite numbers when a digit is changed. In 2007 Jens Kruse Andersen found the 1000-digit weakly prime (17−17)/99 + 21686652. This is the largest known weakly prime number

title: Prime number

text: correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1. By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit". Writing a number as

title: Prime number

text: method can detect both single-digit errors and transpositions of adjacent digits. Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than formula_233. Prime numbers are also used in pseudorandom number generators including linear congruential generators and the Mersenne Twister. Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including abstract algebra and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line, or so that every triangle formed by three of

title: Prime number

text: factorizations are based on Euclid's lemma: If formula_20 is a prime number and formula_20 divides a product formula_33 of integers formula_34 and formula_35, then formula_20 divides formula_34 or formula_20 divides formula_35 (or both). Conversely, if a number formula_20 has the property that when it divides a product it always divides at least one factor of the product, then formula_20 must be prime. There are infinitely many prime numbers. Another way of saying this is that the sequence of prime numbers never ends. This statement is referred to as "Euclid's theorem" in honor of the ancient Greek mathematician Euclid, since the

title: Weakly prime number

text: . There are infinitely many weakly prime numbers in any base. Furthermore, for any fixed base there is a positive proportion of such primes. The smallest base "b" weakly primes for "b" = 1 to 16 are: Weakly prime number In number theory, a prime number is called weakly prime if it becomes not prime when any one of its digits is changed to every single other digit. Decimal digits are usually assumed. The first weakly prime numbers are: For the first of these, each of the 54 numbers 094001, 194001, 394001, ..., 294009 are composite. A weakly prime base-"b"

title: Prime number

text: hash functions by choosing random linear functions modulo large prime numbers. Carter and Wegman generalized this method to formula_64-independent hashing by using higher-degree polynomials, again modulo large primes. As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that the probe sequence covers the whole table. Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this

title: Prime number

text: concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers. The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by Chebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's

title: Prime number theorem

text: of where exceeds is probably around ; see the article on Skewes' number for more details. (On the other hand, the offset logarithmic integral is smaller than already for ; indeed, , while .) In the first half of the twentieth century, some mathematicians (notably G. H. Hardy) believed that there exists a hierarchy of proof methods in mathematics depending on what sorts of numbers (integers, reals, complex) a proof requires, and that the prime number theorem (PNT) is a "deep" theorem by virtue of requiring complex analysis. This belief was somewhat shaken by a proof of the PNT based

title: Prime number

text: to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers. The spectrum of a ring is a geometric space whose points are the prime ideals of the ring. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. These concepts can even assist with in number-theoretic questions solely

title: Prime number

text: formula_69. Prime gaps can be generalized to prime formula_64-tuples, patterns in the differences between more than two prime numbers. Their infinitude and density are the subject of the first Hardy–Littlewood conjecture, which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem. Analytic number theory studies number theory through the lens of continuous functions, limits, infinite series, and the related mathematics of the infinite and infinitesimal. This area of study began with Leonhard Euler and his first major result, the solution to the Basel

title: Prime number

text: this decreases exponentially with the number of tests, it provides high confidence (although not certainty) that a number that passes the repeated test is prime. On the other hand, if the test ever fails, then the number is certainly composite. A composite number that passes such a test is called a pseudoprime. In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. For instance, this is true of trial division. The algorithms with guaranteed-correct output include both deterministic (non-random) algorithms,

title: Prime number theorem

text: limit of the "difference" of the two functions as increases without bound. Instead, the theorem states that approximates in the sense that the relative error of this approximation approaches 0 as increases without bound. The prime number theorem is equivalent to the statement that the th prime number satisfies the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as increases without bound. For example, the th prime number is , and ()log() rounds to , a relative error of about 6.4%. The prime number theorem is also equivalent to where and are the first and

title: Prime number

text: theorem on primes in arithmetic progressions as a special case. In the theory of finite groups the Sylow theorems imply that, if a power of a prime number formula_195 divides the order of a group, then it has a subgroup of order formula_195. By Lagrange's theorem, any group of prime order is a cyclic group, and by the Burnside theorem any group whose order is divisible by only two primes is solvable. 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

title: Prime number

text: is prime. The most basic primality testing routine, trial division, is too slow to be useful for large numbers. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for numbers of special types. Most primality tests only tell whether their argument is prime or not. Routines that also provide a prime factor of composite arguments (or all of its prime factors) are called factorization algorithms. Prime numbers are also used in computing for checksums, hash tables, and pseudorandom number generators. The most basic method of checking the primality of a given integer

title: Prime number

text: one that is prime; when doing this, a faster probabilistic test can quickly eliminate most composite numbers before a guaranteed-correct algorithm is used to verify that the remaining numbers are prime. The following table lists some of these tests. Their running time is given in terms of formula_1, the number to be tested and, for probabilistic algorithms, the number formula_64 of tests performed. Moreover, formula_219 is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that each time bound should be multiplied by a constant factor to convert it from

title: Prime number

text: large threshold, is described by the prime number theorem, but no efficient formula for the formula_1-th prime is known. Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials with relatively prime integers formula_34 and formula_35 take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same formula_35 have approximately the same proportions of primes. Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists

title: Prime number theorem

text: Prime number theorem In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function and is the natural logarithm of . This means that for large enough

title: Prime number

text: prime. In the 19th century many mathematicians still considered 1 to be prime, and lists of primes that included 1 continued to be published as recently as 1956. If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1. Similarly, the sieve of Eratosthenes would not work

title: Prime number

text: so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression can have more than one prime only when its remainder formula_34 and modulus formula_107 are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The Green–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes. Euler noted that the function yields prime numbers for formula_109, although composite numbers appear among its later values. The search for an explanation for this phenomenon led

title: Prime number theorem

text: column, , is the average prime gap below . The value for was originally computed assuming the Riemann hypothesis; it has since been verified unconditionally. There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem. To state it precisely, let be the finite field with elements, for some fixed , and let be the number of monic "irreducible" polynomials over whose degree is equal to . That is, we are looking at polynomials with

title: Prime number theorem

text: Dusart proved: The proof by de la Vallée Poussin implies the following. For every , there is an such that for all , As a consequence of the prime number theorem, one gets an asymptotic expression for the th prime number, denoted by : A better approximation is Again considering the th prime number , this gives an estimate of ; the first 5 digits match and relative error is about 0.00005%. Rosser's theorem states that This can be improved by the following pair of bounds: The table compares exact values of to the two approximations and . The last

title: Prime number

text: outside of mathematics with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G.H. Hardy prided themselves on doing work that had absolutely no military significance. This vision of the purity of number theory 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. These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining whether a given number

title: Prime number

text: values, and places on the rational numbers. The local-global principle allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory. A commutative ring is an algebraic structure where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, "prime elements" and "irreducible elements". An element formula_20 of a ring formula_172 is called prime if it is nonzero, has no multiplicative inverse (that is, it is

title: Prime number

text: is also a sufficient condition for formula_20 to be prime. Wilson's theorem says that an integer formula_145 is prime if and only if the factorial formula_146 is congruent to formula_147 mod formula_20. For a composite this cannot hold, since one of its factors divides both and formula_149, and so formula_150 is impossible. The formula_20-adic order formula_152 of an integer formula_1 is the number of copies of formula_20 in the prime factorization of formula_1. The same concept can be extended from integers to rational numbers by defining the formula_20-adic order of a fraction formula_157 to be formula_158. The formula_20-adic absolute value

title: Prime number

text: prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's "Elements" (circa 300 BC) proves the infinitude of primes and the fundamental theorem of arithmetic, and shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers formula_1 that evenly divide formula_17. Alhazen also conjectured that all even perfect numbers come from Euclid's construction using Mersenne

title: Prime number

text: each of these factors, but formula_45 has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of formula_45 can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes. The numbers formed by adding one to the products of the smallest primes are called Euclid numbers. The first five of them are prime, but the sixth, is a composite number. There is no known efficient formula for primes. For example, there is no non-constant polynomial, even

title: Prime number

text: grow to infinity as formula_1 goes to infinity (see the Basel problem). In this sense, prime numbers occur more often than squares of natural numbers, although both sets are infinite. Brun's theorem states that the sum of the reciprocals of twin primes, is finite. Because of Brun's theorem, it is not possible to use Euler's method to solve the twin prime conjecture, that there exist infinitely many twin primes. The prime counting function formula_88 is defined as the number of primes not greater than formula_1. For example, formula_90, since there are five primes less than or equal to 11. Methods

title: Prime number

text: a quadratic polynomial that (for integer arguments) is prime infinitely often. Euler's proof that there are infinitely many primes considers the sums of reciprocals of primes, Euler showed that, for any arbitrary real number formula_22, there exists a prime formula_20 for which this sum is bigger than formula_22. This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every formula_22. The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum does not

title: Prime number

text: to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy-Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values. The Ulam spiral arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted. Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than

title: Prime number

text: arts and literature. They have also been used in evolutionary biology to explain the life cycles of cicadas. Fermat primes are primes of the form with formula_64 a natural number. They are named after Pierre de Fermat, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, but formula_236 is composite and so are all other Fermat numbers that have been verified as of 2017. A regular formula_1-gon is constructible using straightedge and compass if and only if the odd prime factors of formula_1 (if any)

title: 19 (number)

text: 19 (number) 19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number. In English speech, the numbers 19 and 90 are sometimes confused, as they sound very similar. 19 is the 8th prime number. 19 is the seventh Mersenne prime exponent. 19 is the maximum number of fourth powers needed to sum up to any natural number. It is the fourth value of g(k). 19 is the lowest prime centered triangular number, a centered hexagonal number and a Heegner number. The only non-trivial normal magic hexagon contains 19 hexagons (the other being 1). In

title: Lowest common denominator

text: appeal to the largest possible number of people. Lowest common denominator In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest common multiple. The product of the denominators is always a common denominator, as in: but it is not always the lowest common denominator, as in: Here, 36 is the least common multiple of 12

title: Prime number

text: examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants formula_51 and formula_52 such that are prime for any natural number formula_1 in the first formula, and any number of exponents in the second formula. Here formula_55 represents the floor function, the largest integer less than or equal to the number in question. However, these are not useful for generating primes, as the primes must be generated first in order to compute the values of formula_56 or formula_52. Many conjectures revolving about primes have been posed. Often having an elementary

title: Prime number theorem

text: the notion that the "density" of primes around should be . This function is related to the logarithm by the asymptotic expansion So, the prime number theorem can also be written as . In fact, in another paper in 1899 de la Vallée Poussin proved that for some positive constant , where is the big notation. This has been improved to Because of the connection between the Riemann zeta function and , the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than

title: Prime number

text: and the RSA cryptosystem were invented, using prime numbers as their basis. The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form. The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) on long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size. Most early Greeks did not even consider 1 to be a number, so they could not consider its primality. A few mathematicians from this time also considered

title: Prime number

text: corresponds to "congruence" in modular arithmetic: formula_22 and formula_121 are congruent (written formula_122 mod formula_1) when they have the same remainder after division by formula_1. However, in this system of numbers, division by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number formula_125 as modulus, division by formula_28 is possible: formula_127, because clearing denominators by multiplying both sides by formula_28 gives the valid formula formula_129. However, with the composite modulus formula_130, division by formula_28 is impossible. There is no valid solution to formula_132: clearing denominators by multiplying by formula_28 causes

title: Prime number

text: first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an analytical proof by Euler, Goldbach's proof based on Fermat numbers, Furstenberg's proof using general topology, and Kummer's elegant proof. Euclid's proof shows that every finite list of primes is incomplete. The key idea is to multiply together the primes in any given list and add formula_42. If the list consists of the primes formula_43, this gives the number By the fundamental theorem, formula_45 has a prime factorization with one or more prime factors. formula_45 is evenly divisible by

title: Prime number

text: a special form, including the special number field sieve. the largest number known to have been factored by a general-purpose algorithm is RSA-768, which has 232 decimal digits (768 bits) and is the product of two large primes. Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer. However, current technology can only run this algorithm for very small numbers. the largest number that has been factored by a quantum computer running Shor's algorithm is 21. Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers

title: Prime number

text: listing all of the primes or prime factorizations up to a given limit were commonly printed. The oldest method for generating a list of primes is called the sieve of Eratosthenes. The animation shows an optimized variant of this method. Another more efficient sieving method for the same problem is the sieve of Atkin. In advanced mathematics, sieve theory applies similar methods to other problems. Some of the fastest modern tests for whether an arbitrary given number formula_1 is prime are probabilistic (or Monte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer. For

title: Prime number theorem

text: remaining terms. The "Riemann hypothesis" statement depends on the fact that the largest proper divisor of can be no larger than . Prime number theorem In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such

title: Prime number

text: problem. The problem asked for the value of the infinite sum formula_71 which today can be recognized as the value formula_72 of the Riemann zeta function. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the Riemann hypothesis. Euler showed that formula_73. The reciprocal of this number, formula_74, is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common). The distribution of primes in the large, such as the question how many primes are smaller than a given,

title: Prime number theorem

text: we have in particular empirically the primes congruent to 3 are more numerous and are nearly always ahead in this "prime number race"; the first reversal occurs at . However Littlewood showed in 1914 that there are infinitely many sign changes for the function so the lead in the race switches back and forth infinitely many times. The phenomenon that is ahead most of the time is called Chebyshev's bias. The prime number race generalizes to other moduli and is the subject of much research; Pál Turán asked whether it is always the case that and change places when and

title: Prime number theorem

text: of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of as goes to infinity exists at all, then it is necessarily equal to one. He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near 1, for all sufficiently large . Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for were strong enough for him to prove Bertrand's postulate that there exists a prime number between and for any integer . An important paper concerning the

title: Prime number

text: formula_18, and Marin Mersenne studied the Mersenne primes, prime numbers of the form formula_19 with formula_20 itself a prime. Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes formula_21. At the start of the 19th century, Legendre and Gauss

title: Prime number

text: theorem of arithmetic". This theorem states that every integer larger than 1 can be written as a product of one or more primes. More strongly, this product is unique in the sense that any two prime factorizations of the same number will have the same numbers of copies of the same primes, although their ordering may differ. So, although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes can thus be considered the "basic building blocks" of the natural numbers. Some proofs of the uniqueness of prime

title: Prime number

text: numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877), Proth's theorem (around 1878), the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test. Since 1951 all the largest known primes have been found using these tests on computers. The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects. The idea that prime numbers had few applications outside of pure mathematics was shattered in the 1970s when public-key cryptography

title: Prime number

text: the left-hand side to become formula_134 while the right-hand side becomes either formula_135 or formula_28. In the terminology of abstract algebra, the ability to perform division means that modular arithmetic modulo a prime number forms a field or, more specifically, a finite field, while other moduli only give a ring but not a field. Several theorems about primes can be formulated using modular arithmetic. For instance, Fermat's little theorem states that if formula_137 (mod formula_20), then formula_139 (mod formula_20). Summing this over all choices of formula_34 gives the equation valid whenever formula_20 is prime. Giuga's conjecture says that this equation

title: Zeisel number

text: additionally has prime factors with the relationship described above, so a name like Brown-Zeisel Numbers might be more appropriate. Hardy–Ramanujan's number 1729 is also a Zeisel number. Zeisel number A Zeisel number, named after Helmut Zeisel, is a square-free integer "k" with at least three prime factors which fall into the pattern where "a" and "b" are some integer constants and "x" is the index number of each prime factor in the factorization, sorted from lowest to highest. For the purpose of determining Zeisel numbers, formula_2. The first few Zeisel numbers are To give an example, 1729 is a Zeisel

title: Lowest common denominator

text: Lowest common denominator In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest common multiple. The product of the denominators is always a common denominator, as in: but it is not always the lowest common denominator, as in: Here, 36 is the least common multiple of 12 and 18. Their product, 216, is also a

title: Prime number

text: conjectured that as formula_22 tends to infinity, the number of primes up to formula_22 is asymptotic to formula_24, where formula_25 is the natural logarithm of formula_22. Ideas of Riemann in his 1859 paper on the zeta-function sketched an outline for proving this. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem. Another important 19th-century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes. Many mathematicians have worked on primality tests for

title: 261 (number)

text: was once the lowest number not to have its own Wikipedia page, this making it a candidate for the lowest uninteresting Number according to the definition given by Alex Bellos. , the smallest natural number without its own Wikipedia page is 262, and the smallest prime number without its own Wikipedia page is 283. 261 may refer to... The years 261 AD or 261 BC. The steam locomotive Milwaukee Road 261. Alaska Airlines Flight 261, which crashed in the Pacific Ocean near Anacapa Island on January 31, 2000. 261 (number) 261 (two hundred [and] sixty-one) is a natural number proceeded

title: Prime number

text: dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters formula_1 and formula_64. In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. For example, the Lucas–Lehmer primality test can determine whether a Mersenne number (one less than a power of two) is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test. This is why since 1992 () the largest "known"

title: Wolstenholme prime

text: primes therefore form a subset of the irregular primes. A Wolstenholme prime is a prime "p" such that ("p", "p"–3) is an irregular pair. A Wolstenholme prime is a prime "p" such that i.e. the numerator of the harmonic number formula_4 expressed in lowest terms is divisible by "p". The search for Wolstenholme primes began in the 1960s and continued over the following decades, with the latest results published in 2007. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time. The 1964 discovery was later independently confirmed in the 1970s. This

title: Lowest of the Low (book)

text: Lowest of the Low (book) Lowest of the Low or Lowest of the Low: the Turkish worker in West Germany is the English translation of German journalist Günter Wallraff’s book Ganz unten (“At the bottom”), originally published in Germany in 1985. The book describes his undercover experiment as a Turkish migrant worker in West Germany over the span of two years. Wallraff assumes the identity of a Turkish man named Ali Sigirlioğlu. Wallraff opts to disguise himself for his experiment. He alters his appearance by wearing contact lenses and a wig to make himself appear younger, and additionally adopts what

title: Prime number

text: asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of formula_22 for intervals near a number formula_22). Modular arithmetic modifies usual arithmetic by only using the numbers formula_117, for a natural number formula_1 called the modulus. Any other natural number can be mapped into this system by replacing it by its remainder after division by formula_1. Modular sums, differences and products are calculated by performing the same replacement by the remainder on the result of the usual sum, difference, or product of integers. Equality of integers

title: Prime number theorem

text: distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function originates. Extending the ideas of Riemann, two proofs of the asymptotic law of

title: Zeisel number

text: Zeisel number A Zeisel number, named after Helmut Zeisel, is a square-free integer "k" with at least three prime factors which fall into the pattern where "a" and "b" are some integer constants and "x" is the index number of each prime factor in the factorization, sorted from lowest to highest. For the purpose of determining Zeisel numbers, formula_2. The first few Zeisel numbers are To give an example, 1729 is a Zeisel number with the constants "a" = 1 and "b" = 6, its factors being 7, 13 and 19, falling into the pattern 1729 is an example for

title: What Is Love

text: What Is Love "What Is Love" is a song recorded by Trinidadian-German Eurodance artist Haddaway for his debut album, "The Album". The song is recognizable by its refrain "What is love? Baby don't hurt me, don't hurt me... no more." It was a massive hit in Europe, becoming a number-one hit in at least 13 countries and reaching number two in Germany, Sweden, and the United Kingdom. Outside Europe, the song was moderately successful, reaching number 11 in the United States, number 12 in Australia, number 17 in Canada, and number 48 in New Zealand. "What Is Love" was written

title: Prime number

text: the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. The prime decomposition of 3-manifolds is another example of this type. Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics, and have been used metaphorically in the

title: Prime number

text: is a consequence of Fermat's theorem on sums of two squares, which states that an odd prime formula_20 is expressible as the sum of two squares, formula_187, and therefore factorizable as formula_188, exactly when formula_20 is 1 mod 4. Not every ring is a unique factorization domain. For instance, in the ring of numbers formula_190 (for integers formula_34 and formula_35) the number formula_193 has two factorizations formula_194, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion

title: Prime number theorem

text: the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function is non-zero for all complex values of the variable that have the form with . During the 20th century, the theorem of Hadamard and de la Vallée Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdős (1949). While

title: Prime number

text: prime has always been a Mersenne prime. It is conjectured that there are infinitely many Mersenne primes. The following table gives the largest known primes of various types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Given a composite integer formula_1, the task of providing one (or all) prime factors is

title: 593 (number)

text: 593 (number) 593 (five hundred [and] ninety-three) is the natural number following 592 and preceding 594. 593 is an odd number. It is a prime number, an example of what Paul Erdős and Ernst G. Straus called a Good prime, or a prime whose square is greater than the product of its neighboring two primes. As such it is part of sequence at the On-Line Encyclopedia of Integer Sequences. Justin Smith calls 593 a right prime because it remains prime after dropping any number of digits from the right: 593, 59, and 5 are all prime. It is a Sophie

title: What Is the What

text: What Is the What What Is the What: The Autobiography of Valentino Achak Deng is a 2006 novel written by Dave Eggers. It is based on the life of Valentino Achak Deng, a Sudanese child refugee who immigrated to the United States under the Lost Boys of Sudan program. It was a finalist for the National Book Award. As a boy, Achak is separated from his family during the Second Sudanese Civil War when the Arab militia, referred to as "murahaleen" (which is Arabic for the deported), wipes out his Dinka village, Marial Bai. During the assault, he loses sight

title: What Is the What

text: That Happens Will Happen Today". Tom Tykwer plans to adapt the novel into a film. In 2009, the novel received the Prix Médicis étranger in France. What Is the What What Is the What: The Autobiography of Valentino Achak Deng is a 2006 novel written by Dave Eggers. It is based on the life of Valentino Achak Deng, a Sudanese child refugee who immigrated to the United States under the Lost Boys of Sudan program. It was a finalist for the National Book Award. As a boy, Achak is separated from his family during the Second Sudanese Civil War when

title: Forte number

text: in the set, and an "off" bit represents one that is not. The prime of a set is the rotational transposition with the lowest numeric value. The result of this simple algorithm produces the same results as Rahn, and because it employs a binary representation it has distinct advantages when manipulating primes using computation. Elliott Carter had earlier (1960-67) produced a numbered listing of pitch class sets, or "chords", as Carter referred to them, for his own use. Forte number In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of

title: Landau prime ideal theorem

text: a simple pole with residue −1 at "s" = 1. As with the Prime Number Theorem, a more precise estimate may be given in terms of the logarithmic integral function. The number of prime ideals of norm ≤ "X" is where "c" is a constant depending on "K". Landau prime ideal theorem In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field "K", with norm at most "X". What to expect can be seen already for

title: Lowest common denominator

text: numerators of fractions with common denominators can simply be added, such that formula_7 and that formula_8, since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what formula_9 equals, or whether formula_10 is greater than or less than formula_11. Any common denominator will do, but usually the lowest common denominator is desirable because it makes the rest of the calculation as simple as possible. The expression "lowest common denominator" is used to describe (usually in a disapproving manner) a rule, proposal, opinion, or media that is deliberately simplified so as to

title: The Lowest

text: The Lowest The Lowest is an album by bassist and tubist Red Callender, recorded for the MetroJazz label in 1958. The Allmusic review by Ken Dryden states: "These sessions feature quite a few of the rising stars of the West Coast jazz scene... As a bassist, Callender was very much in demand and displays his immense talent in a rather subdued way. He sets aside his bass to play tuba on several tracks, proving himself as a convincing soloist on an instrument almost always relegated to rhythm, in addition to being out of favor once the bass came into jazz

title: The Lowest

text: ensembles". All compositions by Red Callender except where noted The Lowest The Lowest is an album by bassist and tubist Red Callender, recorded for the MetroJazz label in 1958. The Allmusic review by Ken Dryden states: "These sessions feature quite a few of the rising stars of the West Coast jazz scene... As a bassist, Callender was very much in demand and displays his immense talent in a rather subdued way. He sets aside his bass to play tuba on several tracks, proving himself as a convincing soloist on an instrument almost always relegated to rhythm, in addition to being

title: Prime number theorem

text: , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (). In other words, the average gap between consecutive prime numbers among the first integers is roughly . Let be the prime-counting

title: Landau prime ideal theorem

text: Landau prime ideal theorem In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field "K", with norm at most "X". What to expect can be seen already for the Gaussian integers. There for any prime number "p" of the form 4"n" + 1, "p" factors as a product of two Gaussian primes of norm "p". Primes of the form 4"n" + 3 remain prime, giving a Gaussian prime of norm "p". Therefore, we should estimate where