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Euclid Number
In mathematics, Euclid numbers are integers of the form , where ''p''''n''# is the ''n''th primorial, i.e. the product of the first ''n'' prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers. Examples For example, the first three primes are 2, 3, 5; their product is 30, and the corresponding Euclid number is 31. The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, ... . History It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of prime numbers relied on these numbers. Euclid did not begin with the assumption that the set of all primes is finite. Rather, he said: consider any finite set of primes (he did not assume that it contained only the first ''n'' primes, e.g. it could have been ) and reasoned from there to the conclusion that at least one prime exists that is not in that set. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
211 (number)
211 (two hundred ndeleven) is the natural number following 210 and preceding 212. It is also a prime number. In mathematics 211 is an odd number. 211 is a primorial prime, sum of three consecutive primes (67 + 71 + 73), Chen prime, centered decagonal prime, and self prime. 211 is the smallest prime separated by 10 or more from the nearest primes (199 and 223). It is thus a balanced prime and an ''isolated prime''. 211 is a repdigit in base 14 (111). Multiplying its digits, it is still a prime (2), and adding its digits, it is square (4). Rearranging its digits, 211 becomes 121, which also is a square. Adding any two of its digits will be prime (2 or 3). 211 is a super-prime. In science and technology 2-1-1 is special abbreviated telephone number reserved in Canada and the United States as an easy-to-remember three-digit telephone number. It is meant to provide quick information and referrals to health and human service organizations for both services from charities and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of 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, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either 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 n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. 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 pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid–Mullin Sequence
The Euclid–Mullin sequence is an infinite sequence of distinct prime numbers, in which each element is the least prime factor of one plus the product of all earlier elements. They are named after the ancient Greek mathematician Euclid, because their definition relies on an idea in Euclid's proof that there are infinitely many primes, and after Albert A. Mullin, who asked about the sequence in 1963. The first 51 elements of the sequence are :2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357, 87991098722552272708281251793312351581099392851768893748012603709343, 107, 127, 3313, 227432689108589532754984915075774848386671439568260420754414940780761245893, 59, 31, 211... These are the only known elements . Finding the next one requires finding the leas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
209 (number)
209 (two hundred ndnine) is the natural number following 208 and preceding 210. In mathematics *There are 209 spanning trees in a 2 × 5 grid graph, 209 partial permutations on four elements, and 209 distinct undirected simple graphs on 7 or fewer unlabeled vertices. *209 is the smallest number with six representations as a sum of three positive squares. These representations are: *:209 . :By Legendre's three-square theorem, all numbers congruent to 1, 2, 3, 5, or 6 mod 8 have representations as sums of three squares, but this theorem does not explain the high number of such representations for 209. *, one less than the product of the first four prime numbers. Therefore, 209 is a Euclid number of the second kind, also called a Kummer number. One standard proof of Euclid's theorem that there are infinitely many primes uses the Kummer numbers, by observing that the prime factors of any Kummer number must be distinct from the primes in its product formula as a Kummer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squarefree Number
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial Prime
In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes). Primality tests show that : ''pn''# − 1 is prime for ''n'' = 2, 3, 5, 6, 13, 24, ... : ''pn''# + 1 is prime for ''n'' = 0, 1, 2, 3, 4, 5, 11, ... The first term of the second sequence is 0 because ''p''0# = 1 is the empty product, and thus ''p''0# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, because ''p''1# = 2, and 2 − 1 = 1 is not prime. The first few primorial primes are : 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 , the largest known primorial prime (of the form ''p''''n''# − 1) is 3267113# − 1 (''n'' = 234,725) with 1,418,398 digits, found by the PrimeGrid project. , the largest known prime of the form ''p''''n''# + 1 is 392113# + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Intelligencer
''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quarterly with a subset of open access articles. Springer also cross-publishes some of the articles in ''Scientific American''. Karen Parshall and Sergei Tabachnikov are currently the co-editors-in-chief. History The journal was started informally in 1971 by Walter Kaufman-Buehler, Alice Peters and Klaus Peters. "Intelligencer" was chosen by Kaufman-Buehler as a word that would appear slightly old-fashioned. An exploration of mathematically themed stamps, written by Robin Wilson, became one of its earliest columns. In 1978, the founders appointed Bruce Chandler and Harold "Ed" Edwards Jr. to serve jointly in the role of editor-in-chief. Prior to 1978, articles of the ''Intelligencer'' were not contained in regular volumes and were sent out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
31 (number)
31 (thirty-one) is the natural number following thirty, 30 and preceding 32 (number), 32. It is a prime number. In mathematics 31 is the 11th prime number. It is a superprime and a Self number#Self primes, self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is a lucky prime and a happy number; two properties it shares with 13 (number), 13, which is its dual emirp and permutable prime. 31 is also a primorial prime, like its twin prime, 29 (number), 29. 31 is the number of regular polygons with an odd number of sides that are known to be constructible polygon, constructible with compass and straightedge, from combinations of known Fermat primes of the form 22''n'' + 1. 31 is the third Mersenne prime of the form 2''n'' − 1. It is also the eighth Mersenne prime exponent, specifically for the number 2,147,483,647, which is the maximum positive value for a 32-bit Integer (computer science), signed binary integer in computing. After 3, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7 (number)
7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube. As an early prime number in the series of positive integers, the number seven has greatly symbolic associations in religion, mythology, superstition and philosophy. The seven Classical planets resulted in seven being the number of days in a week. It is often considered lucky in Western culture and is often seen as highly symbolic. Unlike Western culture, in Vietnamese culture, the number seven is sometimes considered unlucky. It is the first natural number whose pronunciation contains more than one syllable. Evolution of the Arabic digit In the beginning, Indians wrote 7 more or less in one stroke as a curve that looks like an uppercase vertically inverted. The western Ghubar Arabs' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the digit more rectilinear. The eastern Arabs developed the digit fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
