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27th
27 (twenty-seven; Roman numeral XXVII) is the natural number following 26 and preceding 28. In mathematics * Twenty-seven is a cube of 3: 3^3=3\times 3\times 3. 27 is also 23 (see tetration). There are exactly 27 straight lines on a smooth cubic surface, which give a basis of the fundamental representation of the E6 Lie algebra. 27 is also a decagonal number. * In decimal, it is the first composite number not divisible by any of its digits. * It is the radix (base) of the septemvigesimal positional numeral system. * 27 is the only positive integer that is 3 times the sum of its digits. * In a prime reciprocal magic square of the multiples of , the magic constant is 27. * In the Collatz conjecture (aka the "3n+1 conjecture"), a starting value of 27 requires 111 steps to reach 1, more than any number smaller than it. * The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional. * In decim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
26 (number)
26 (twenty-six) is the natural number following 25 and preceding 27. In mathematics *26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1). *26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise connections. *There are 26 sporadic groups. *The 26-dimensional Lorentzian unimodular lattice II25,1 plays a significant role in sphere packing problems and the classification of finite simple groups. In particular, the Leech lattice is obtained in a simple way as a subquotient. *26 is the smallest number that is both a nontotient and a noncototient number. *There are 26 faces of a rhombicuboctahedron. *When a 3 × 3 × 3 cube is made of 27 unit cubes, 26 of them are viewable as the exterior layer. Also a 26 sided polygon is called an icosihexagon. *φ(26) = φ(σ(26)). Properties of its positional representation in certain radixes Twenty-six is a repdigit in base three (2223) and in base 12 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collatz Conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. It is named after mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. It is also known as the problem, the conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem. The sequence of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dark Matter
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not absorb, reflect, or emit electromagnetic radiation and is, therefore, difficult to detect. Various astrophysical observationsincluding gravitational effects which cannot be explained by currently accepted theories of gravity unless more matter is present than can be seenimply dark matter's presence. For this reason, most experts think that dark matter is abundant in the universe and has had a strong influence on its structure and evolution. The primary evidence for dark matter comes from calculations showing that many galaxies would behave quite differently if they did not contain a large amount of unseen matter. Some galaxies would not have formed at all and others would not move as they currently do. Other lines of evidence include observa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aluminum
Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. It has a great affinity towards oxygen, and forms a protective layer of oxide on the surface when exposed to air. Aluminium visually resembles silver, both in its color and in its great ability to reflect light. It is soft, non-magnetic and ductile. It has one stable isotope, 27Al; this isotope is very common, making aluminium the twelfth most common element in the Universe. The radioactivity of 26Al is used in radiodating. Chemically, aluminium is a post-transition metal in the boron group; as is common for the group, aluminium forms compounds primarily in the +3 oxidation state. The aluminium cation Al3+ is small and highly charged; as such, it is polarizing, and bonds aluminium forms tend towards covalency. The strong affinity towards ox ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobalt
Cobalt is a chemical element with the symbol Co and atomic number 27. As with nickel, cobalt is found in the Earth's crust only in a chemically combined form, save for small deposits found in alloys of natural meteoric iron. The free element, produced by reductive smelting, is a hard, lustrous, silver-gray metal. Cobalt-based blue pigments ( cobalt blue) have been used since ancient times for jewelry and paints, and to impart a distinctive blue tint to glass, but the color was for a long time thought to be due to the known metal bismuth. Miners had long used the name ''kobold ore'' (German for ''goblin ore'') for some of the blue-pigment-producing minerals; they were so named because they were poor in known metals, and gave poisonous arsenic-containing fumes when smelted. In 1735, such ores were found to be reducible to a new metal (the first discovered since ancient times), and this was ultimately named for the ''kobold''. Today, some cobalt is produced specifically from one of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every atom of that element. The atomic number can be used to uniquely identify ordinary chemical elements. In an ordinary uncharged atom, the atomic number is also equal to the number of electrons. For an ordinary atom, the sum of the atomic number ''Z'' and the neutron number ''N'' gives the atom's atomic mass number ''A''. Since protons and neutrons have approximately the same mass (and the mass of the electrons is negligible for many purposes) and the mass defect of the nucleon binding is always small compared to the nucleon mass, the atomic mass of any atom, when expressed in unified atomic mass units (making a quantity called the "relative isotopic mass"), is within 1% of the whole number ''A''. Atoms with the same atomic number but dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Numbers
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] |
Senary
A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like decimal, it is a semiprime, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to senary. In turn, the senary logic refers to an extension of Jan Łukasiewicz's and Stephen Cole Kleene's ternary logic systems adjusted to explain the logic of statistical tests and missing data patterns in sciences using empirical methods. Formal definition The standard set of digits in senary is given by \mathcal_6 = \lbrace 0, 1, 2, 3, 4, 5\rbrace, with a linear order 0 < 1 < 2 < 3 < 4 < 5. Let be the |
Tits Group
In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order : 211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group. History and properties The Ree groups 2''F''4(22''n''+1) were constructed by , who showed that they are simple if ''n'' ≥ 1. The first member of this series 2''F''4(2) is not simple. It was studied by who showed that it is almost simple, its derived subgroup 2''F''4(2)′ of index 2 being a new simple group, now called the Tits group. The group 2''F''4(2) is a group of Lie type and has a BN pair, but the Tits group itself does not have a BN pair. Because the Tits group is not strictly a group of Lie type, it is sometimes regarded as a 27th sporadic group.For instance, by the ATLAS of Finite Groups and itweb-based descendant/ref> The Schur multiplier of the Tits group is trivial and its outer automorphism group has orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sporadic Group
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups. The monster group is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it. Names Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harshad Number
In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit ' (joy) + ' (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. Definition Stated mathematically, let be a positive integer with digits when written in base , and let the digits be a_i (i = 0, 1, \ldots, m-1). (It follows that a_i must be either zero or a positive integer up to .) can be expressed as :X=\sum_^ a_i n^i. is a harshad number in base if: :X \equiv 0 \bmod . A number which is a harshad number in every number base is called an all-harshad number, or an all-Niven number. There are only four all-harshad numbers: 1, 2, 4, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smith Number
In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the given number base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed. Smith numbers were named by Albert Wilansky of Lehigh University, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith: : 4937775 = 31 52 658371 while : 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 · 1 + 5 · 2 + (6 + 5 + 8 + 3 + 7) · 1 = 42 in base 10.Sándor & Crstici (2004) p.383 Mathematical definition Let n be a natural number. For base b > 1, let the function F_(n) be the digit sum of n in base b. A natural number n has the integer factorisation : n = \prod_ p^ and is a Smith number if : F_b(n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
