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76 (number)
76 (seventy-six) is the natural number following 75 and preceding 77. In mathematics 76 is: * a composite number; a square-prime, of the form (''p''2, q) where q is a higher prime. It is the ninth of this general form and the seventh of the form (22.q). * a Lucas number. * a telephone or involution number, the number of different ways of connecting 6 points with pairwise connections. * a nontotient. * a 14-gonal number. * a centered pentagonal number. * an Erdős–Woods number since it is possible to find sequences of 76 consecutive integers such that each inner member shares a factor with either the first or the last member. * with an aliquot sum of 64; within an aliquot sequence of two composite numbers (76, 64, 63, 1,0) to the Prime in the 63-aliquot tree. * an automorphic number in base 10. It is one of two 2-digit numbers whose square, 5,776, and higher powers, end in the same two digits. The other is . There are 76 unique compact uniform hyperbolic honeycombs in the ... [...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] |
64 (number)
64 (sixty-four) is the natural number following 63 (number), 63 and preceding 65 (number), 65. Mathematics Sixty-four is the square of 8 (number), 8, the cube of 4 (number), 4, and the sixth power of 2 (number), 2. It is the seventeenth interprime, since it lies midway between the eighteenth and nineteenth prime numbers (61 (number), 61, 67 (number), 67). The aliquot sum of a power of two (2''n'') is always one less than the power of two itself, therefore the aliquot sum of 64 is 63 (number), 63, within an aliquot sequence of two composite members (64, 63 (number), 63, 41 (number), 41, 1 (number), 1, 0) that are rooted in the aliquot tree of the thirteenth prime, 41. 64 is: *the smallest number with exactly seven divisors, *the first whole number (greater than one) that is both a perfect square, and a perfect cube, *the lowest positive power of two that is not adjacent to either a Mersenne prime or a Fermat prime, *the fourth superperfect number — a number such that divisor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wythoff Construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process The method is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope. However, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point. They therefore enclose a spherical triangle on the surface of any sphere centered on that point and repeated reflections produce a multitude of copies of the triangle. If the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a suitable point inside the spherical triangle enclosed by the mirrors, it is possible to ensure that the reflections of that po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Third Dimension
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called ''3-manifolds''. The term may also refer colloquially to a subset of space, a ''three-dimensional region'' (or 3D domain), a ''solid figure''. Technically, a tuple of numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted \R^n, and can be identified to the pair formed by a -dimensional Euclidean space and a Cartesian coordinate system. When , this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics, it serves as a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Hyperbolic Honeycomb
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedron, uniform polyhedral Cell (geometry), cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of Coxeter-Dynkin diagram#Application with uniform polytopes, rings of the Coxeter diagrams for each family. Hyperbolic uniform honeycomb families Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups. Compact uniform honeycomb families The nine compact Coxeter groups are listed here with their Coxeter diagrams, in order of the relative volumes of their Fundamental domain, fundamental simplex domains.Felikson, 2002 These 9 families generate a total of 76 unique uniform honeycombs. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Number
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base b whose square "ends" in the same digits as the number itself. Definition and properties Given a number base b, a natural number n with k digits is an automorphic number if n is a fixed point of the polynomial function f(x) = x^2 over \mathbb/b^k\mathbb, the ring of integers modulo b^k. As the inverse limit of \mathbb/b^k\mathbb is \mathbb_b, the ring of b-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of f(x) = x^2 over \mathbb_b. For example, with b = 10, there are four 10-adic fixed points of f(x) = x^2, the last 10 digits of which are: : \ldots 0000000000 : \ldots 0000000001 : \ldots 8212890625 : \ldots 1787109376 Thus, the automorphic numbers in base 10 are 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 82 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. In mathematics The number 1 is the first natural number after 0. Each natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
63 (number)
63 (sixty-three) is the natural number following 62 (number), 62 and preceding 64 (number), 64. Mathematics 63 is the sum of the first six Exponentiation, powers of 2 (20 + 21 + ... 32 (number), 25). It is the eighth highly cototient number, and the fourth centered octahedral number after 7 and 25 (number), 25. For five unlabeled elements, there are 63 posets. Sixty-three is the seventh ''square-prime'' of the form \, p^ \times q and the second of the form 3^ \times q. It contains a prime aliquot sum of 41 (number), 41, the thirteenth Sequence, indexed prime; and part of the aliquot sequence (63, 41, 1, 0) within the 41-aliquot tree. Zsigmondy's theorem states that where a>b>0 are coprime integers for any integer n \ge 1, there exists a ''primitive prime divisor'' p that divides a^n-b^n and does not divide a^k-b^k for any positive integer k [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliquot Sequence
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Definition and overview The aliquot sequence starting with a positive integer can be defined formally in terms of the sum-of-divisors function or the aliquot sum function in the following way: \begin s_0 &= k \\ pts_n &= s(s_) = \sigma_1(s_) - s_ \quad \text \quad s_ > 0 \\ pts_n &= 0 \quad \text \quad s_ = 0 \\ pts(0) &= \text \end If the condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6. For example, the aliquot sequence of 10 is because: \begin \sigma_1(10) -10 &= 5 + 2 + 1 = 8, \\ pt\sigma_1(8) - 8 &= 4 + 2 + 1 = 7, \\ pt\sigma_1(7) - 7 &= 1, \\ pt\sigma_1(1) - 1 &= 0. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliquot Sum
In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. That is, s(n)=\sum_ d \, . It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. Examples For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. (). The values of for are: :0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ... Characterization of classes of numbers The aliquot sum function can be used to characterize several notable classes of numbers: *1 is the only number whose aliquot sum is 0. *A number is prime if and only if its aliquot sum is 1. *The aliquot sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
75 (number)
75 (seventy-five) is the natural number following 74 (number), 74 and preceding 76 (number), 76. __TOC__ In mathematics 75 is a self number because there is no integer that added up to its own digits adds up to 75. It is the sum of the first five pentagonal numbers, and therefore a pentagonal pyramidal number, as well as a nonagonal number. It is also the fourth ordered Bell number, and a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 7 (number), 7, 5 (number), 5, 12 (number), 12, 17 (number), 17, 29 (number), 29, 46 (number), 46, 75... 75 is the count of the number of weak orderings on a set of four items. Excluding the infinite sets, there are 75 uniform polyhedra in the third dimension, which incorporate Star polyhedron, star polyhedra as well. Inclusive of 7 families of Prism (geometry), prisms and antiprism, antiprisms, there are also 75 uniform polyhedron compound, uniform compound polyhedra. References Integers {{N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Woods Number
In number theory, a positive integer is said to be an Erdős–Woods number if it has the following property: there exists a positive integer such that in the sequence of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints. In other words, is an Erdős–Woods number if there exists a positive integer such that for each integer between and , at least one of the greatest common divisors or is greater than . Examples 16 is an Erdős–Woods number because the 15 numbers between 2184 and each share a prime factor with one of and These 15 numbers and their shared prime factor(s) are: The first Erdős–Woods numbers are Although all of these initial numbers are even, odd Erdős–Woods numbers also exist. They include Prime partitions The Erdős–Woods numbers can be characterized in terms of certain partitions of the prime numbers. A number is an Erdős–Woods number if and only if the prime numbers less than c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
