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116 (number)
116 (one hundred [and] sixteen) is the natural number following 115 (number), 115 and preceding 117 (number), 117. In mathematics 116 is a noncototient, meaning that there is no solution to the equation , where stands for Euler's totient function. 116! + 1 is a factorial prime. There are 116 ternary Lyndon words of length six, and 116 irreducible polynomials of degree six over a three-element field, which form the basis of a free Lie algebra of dimension 116. There are 116 different ways of partitioning the numbers from 1 through 5 into subsets in such a way that, for every ''k'', the union of the first ''k'' subsets is a consecutive sequence of integers. There are 116 different 6×6 Costas arrays.. See also *116 (other) References {{DEFAULTSORT:116 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
115 (number)
115 (one hundred [and] fifteen) is the natural number following 114 (number), 114 and preceding 116 (number), 116. In mathematics 115 has a square number, square sum of divisors: :\sigma(115)=1+5+23+115=144=12^2. There are 115 different rooted trees with exactly eight nodes, 115 inequivalent ways of placing six Rook (chess), rooks on a 6 × 6 chess board in such a way that no two of the rooks attack each other, and 115 solutions to the Map folding, stamp folding problem for a strip of seven stamps. 115 is also a heptagonal pyramidal number. The 115th Woodall number, :115\cdot 2^-1=4\;776\;913\;109\;852\;041\;418\;248\;056\;622\;882\;488\;319, is a prime number. 115 is the sum of the first five heptagonal numbers. See also * 115 (other) References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
117 (number)
117 (one hundred [and] seventeen) is the natural number following 116 (number), 116 and preceding 118 (number), 118. In mathematics 117 is the smallest possible length of the longest edge of an integer Heronian tetrahedron (a tetrahedron whose edge lengths, face areas and volume are all integers). Its other edge lengths are 51, 52, 53, 80 and 84. 117 is a pentagonal number. In other fields 117 can be a substitute for the number 17 (number), 17, which is Heptadecaphobia, considered unlucky in Italy. When Renault exported the Renault 15 and 17, R17 to Italy, it was renamed R117. In the Danish language the number 117 () is often used as a Hyperbole, hyperbolic term to represent an arbitrary but large number. See also * 117 (other) References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncototient
In number theory, a noncototient is a positive integer that cannot be expressed as the difference between a positive integer and the number of coprime integers below it. That is, , where stands for Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ..., has no solution for . The '' cototient'' of is defined as , so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number can be represented as a sum of two distinct primes and , then \begin pq - \varphi(pq) &= pq - (p-1)(q-1) \\ &= p + q - 1 \\ &= n - 1. \end It is expected that every even number larger than 6 is a sum of two disti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial Prime
A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : : 2 (0! + 1 or 1! + 1), 3 (2! + 1), 5 (3! − 1), 7 (3! + 1), 23 (4! − 1), 719 (6! − 1), 5039 (7! − 1), 39916801 (11! + 1), 479001599 (12! − 1), 87178291199 (14! − 1), ... ''n''! − 1 is prime for : :''n'' = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003, 632760, ... (resulting in 28 factorial primes) ''n''! + 1 is prime for : :''n'' = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429, ... (resulting in 24 factorial primes - the prime 2 is repeated) No other factori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyndon Word
In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its rotations. Lyndon words are named after mathematician Roger Lyndon, who investigated them in 1954, calling them standard lexicographic sequences. Anatoly Shirshov introduced Lyndon words in 1953 calling them regular words. Lyndon words are a special case of Hall words; almost all properties of Lyndon words are shared by Hall words. Definitions Several equivalent definitions exist. A k-ary Lyndon word of length n > 0 is an n-character string over an alphabet of size k, and which is the unique minimum element in the lexicographical ordering in the multiset of all its rotations. Being the singularly smallest rotation implies that a Lyndon word differs from any of its non-trivial rotations, and is therefore aperiodic.; . Alternately, a word w is a Lyndon word if and only if it is nonempty and lexicographically stri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as \left(x - \sqrt\right)\left(x + \sqrt\right) if it is considered as a polynomial with real coefficients. One says that the polynomial is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common definitions. Most often, a pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Lie Algebra
In mathematics, a free Lie algebra over a field ''K'' is a Lie algebra generated by a set ''X'', without any imposed relations other than the defining relations of alternating ''K''-bilinearity and the Jacobi identity. Definition The definition of the free Lie algebra generated by a set ''X'' is as follows: : Let ''X'' be a set and i\colon X \to L a morphism of sets ( function) from ''X'' into a Lie algebra ''L''. The Lie algebra ''L'' is called free on ''X'' if i is the universal morphism; that is, if for any Lie algebra ''A'' with a morphism of sets f\colon X \to A, there is a unique Lie algebra morphism g\colon L\to A such that f = g \circ i. Given a set ''X'', one can show that there exists a unique free Lie algebra L(X) generated by ''X''. In the language of category theory, the functor sending a set ''X'' to the Lie algebra generated by ''X'' is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Costas Array
In mathematics, a Costas array can be regarded geometry, geometrically as a set of ''n'' points, each at the center of a square in an ''n''×''n'' square tiling such that each row or column contains only one point, and all of the ''n''(''n'' − 1)/2 displacement (vector), displacement Euclidean vector, vectors between each pair of dots are distinct. This results in an ideal "thumbtack" auto-ambiguity function, making the arrays useful in applications such as sonar and radar. Costas arrays can be regarded as two-dimensional cousins of the one-dimensional Golomb ruler construction, and, as well as being of mathematical interest, have similar applications in experimental design and phased array radar engineering. Costas arrays are named after John P. Costas (engineer), John P. Costas, who first wrote about them in a 1965 technical report. Independently, Edgar Gilbert also wrote about them in the same year, publishing what is now known as the logarithmic Welch metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
116 (other)
116 (''one hundred and sixteen'') may refer to: *116 (number) *AD 116 *116 BC *116 (Devon and Cornwall) Engineer Regiment, Royal Engineers, a military unit *116 (MBTA bus) *116 (New Jersey bus) *116 (hip hop group), a Christian hip hop collective *116 emergency number, see List of emergency telephone numbers ** 116 emergency telephone number in California *116 helplines A harmonised service of social value is a type of freephone service available in the European Union and in some non-EU countries (including the countries in the European Economic Area and United Kingdom), which answers a specific social need, i ... in Europe *Route 116, see list of highways numbered 116 * 116 Sirona, a main-belt asteroid See also * 11/6 (other) * * Livermorium, synthetic chemical element with atomic number 116 {{Numberdis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
