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38-405: Twenty-seven is the cube of 3 , or the 2nd tetration of 3: 2 3 = 3 3 = 3 × 3 × 3 {\displaystyle ^{2}3=3^{3}=3\times 3\times 3} . It is divisible by the number of prime numbers below it ( nine ). The first non-trivial decagonal number is 27. 27 has an aliquot sum of 13 (the sixth prime number) in

76-487: A cube , is a number which is the cube of an integer . The non-negative perfect cubes up to 60 are (sequence A000578 in the OEIS ): Geometrically speaking, a positive integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube , since 3 × 3 × 3 = 27 . The difference between

114-425: A closer examination than they have yet received." (At the time, the other sporadic groups had not been discovered.) The diagram at right is based on Ronan (2006 , p. 247). It does not show the numerous non-sporadic simple subquotients of the sporadic groups. Of the 26 sporadic groups, 20 can be seen inside the monster group as subgroups or quotients of subgroups ( sections ). These twenty have been called

152-594: A method for calculating cube roots in the 1st century CE. Methods for solving cubic equations and extracting cube roots appear in The Nine Chapters on the Mathematical Art , a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE. Sporadic group In the mathematical classification of finite simple groups , there are a number of groups which do not fit into any infinite family. These are called

190-424: A number whose cube is n is called extracting the cube root of n . It determines the side of the cube of a given volume. It is also n raised to the one-third power. The graph of the cube function is known as the cubic parabola . Because the cube function is an odd function, this curve has a center of symmetry at the origin, but no axis of symmetry . A cube number , or a perfect cube , or sometimes just

228-534: A smooth cubic surface , which give a basis of the fundamental representation of Lie algebra E 6 {\displaystyle \mathrm {E_{6}} } . The unique simple formally real Jordan algebra , the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions , is 27-dimensional; its automorphism group is the 52-dimensional exceptional Lie algebra F 4 . {\displaystyle \mathrm {F_{4}} .} There are twenty-seven sporadic groups , if

266-442: Is a cube: with the first one sometimes identified as the mysterious Plato's number . The formula F for finding the sum of n cubes of numbers in arithmetic progression with common difference d and initial cube a , is given by A parametric solution to is known for the special case of d = 1 , or consecutive cubes, as found by Pagliani in 1829. In the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...,

304-511: Is also the fourth perfect totient number — as are all powers of 3 — with its adjacent members 15 and 39 adding to twice 27. A prime reciprocal magic square based on multiples of 1 7 {\displaystyle {\tfrac {1}{7}}} in a 6 × 6 {\displaystyle 6\times 6} square has a magic constant of 27. Including the null-motif, there are 27 distinct hypergraph motifs . There are exactly twenty-seven straight lines on

342-497: Is another solution that is selected. Similarly, for n = 48 , the solution ( x , y , z ) = (-2, -2, 4) is excluded, and this is the solution ( x , y , z ) = (-23, -26, 31) that is selected. The equation x + y = z has no non-trivial (i.e. xyz ≠ 0 ) solutions in integers. In fact, it has none in Eisenstein integers . Both of these statements are also true for the equation x + y = 3 z . The sum of

380-465: Is conjectured that every integer (positive or negative) not congruent to ±4 modulo 9 can be written as a sum of three (positive or negative) cubes with infinitely many ways. For example, 6 = 2 3 + ( − 1 ) 3 + ( − 1 ) 3 {\displaystyle 6=2^{3}+(-1)^{3}+(-1)^{3}} . Integers congruent to ±4 modulo 9 are excluded because they cannot be written as

418-413: Is denoted by a superscript 3, for example 2 = 8 or ( x + 1) . The cube is also the number multiplied by its square : The cube function is the function x ↦ x (often denoted y = x ) that maps a number to its cube. It is an odd function , as The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists of finding

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456-624: Is the Euler–Mascheroni constant ; this hypothesis is true if and only if this inequality holds for every larger n . {\displaystyle n.} In decimal , 27 is the first composite number not divisible by any of its digits, as well as: Also in base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also a multiple of 27. For example, 378, 783, and 837 are all divisible by 27. In senary (base six), one can readily test for divisibility by 43 (decimal 27) by seeing if

494-847: Is the one that is primitive ( gcd( x , y , z ) = 1 ), is not of the form c 3 + ( − c ) 3 + n 3 = n 3 {\displaystyle c^{3}+(-c)^{3}+n^{3}=n^{3}} or ( n + 6 n c 3 ) 3 + ( n − 6 n c 3 ) 3 + ( − 6 n c 2 ) 3 = 2 n 3 {\displaystyle (n+6nc^{3})^{3}+(n-6nc^{3})^{3}+(-6nc^{2})^{3}=2n^{3}} (since they are infinite families of solutions), satisfies 0 ≤ | x | ≤ | y | ≤ | z | , and has minimal values for | z | and | y | (tested in this order). Only primitive solutions are selected since

532-546: Is the second self-locating string after 6 , of only a few known. Twenty-seven is also: Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987), p. 106. Cubic number In arithmetic and algebra , the cube of a number n is its third power , that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression

570-888: Is the sum of three positive rational cubes, and there are rationals that are not the sum of two rational cubes. In real numbers , the cube function preserves the order: larger numbers have larger cubes. In other words, cubes (strictly) monotonically increase . Also, its codomain is the entire real line : the function x ↦ x  : R → R is a surjection (takes all possible values). Only three numbers are equal to their own cubes: −1 , 0 , and 1 . If −1 < x < 0 or 1 < x , then x > x . If x < −1 or 0 < x < 1 , then x < x . All aforementioned properties pertain also to any higher odd power ( x , x , ...) of real numbers. Equalities and inequalities are also true in any ordered ring . Volumes of similar Euclidean solids are related as cubes of their linear sizes. In complex numbers ,

608-744: The non-strict group of Lie type T {\displaystyle \mathrm {T} } (with an irreducible representation that is twice that of F 4 {\displaystyle \mathrm {F_{4}} } in 104 dimensions) is included. In Robin's theorem for the Riemann hypothesis , twenty-seven integers fail to hold σ ( n ) < e γ n log ⁡ log ⁡ n {\displaystyle \sigma (n)<e^{\gamma }n\log \log n} for values n ≤ 5040 , {\displaystyle n\leq 5040,} where γ {\displaystyle \gamma }

646-452: The aliquot sequence ( 27 , 13 , 1 , 0 ) {\displaystyle (27,13,1,0)} of only one composite number, rooted in the 13 -aliquot tree. In the Collatz conjecture (i.e. the 3 n + 1 {\displaystyle 3n+1} problem), a starting value of 27 requires 3 × 37 = 111 steps to reach 1, more than any smaller number. 27

684-483: The happy family by Robert Griess , and can be organized into three generations. M n for n = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M 24 , which is a permutation group on 24 points. All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice : Consists of subgroups which are closely related to

722-433: The sporadic simple groups , or the sporadic finite groups , or just the sporadic groups . A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The mentioned classification theorem states that the list of finite simple groups consists of 18 countably infinite families plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are

760-1001: The 1860s and the other twenty-one were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is: Various constructions for these groups were first compiled in Conway et al. (1985) , including character tables , individual conjugacy classes and lists of maximal subgroup , as well as Schur multipliers and orders of their outer automorphisms . These are also listed online at Wilson et al. (1999) , updated with their group presentations and semi-presentations. The degrees of minimal faithful representation or Brauer characters over fields of characteristic p ≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in Jansen (2005) . A further exception in

798-476: The Monster group M : (This series continues further: the product of M 12 and a group of order 11 is the centralizer of an element of order 11 in M .) The Tits group , if regarded as a sporadic group, would belong in this generation: there is a subgroup S 4 × F 4 (2)′ normalising a 2C subgroup of B , giving rise to a subgroup 2·S 4 × F 4 (2)′ normalising a certain Q 8 subgroup of

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836-511: The Monster. F 4 (2)′ is also a subquotient of the Fischer group Fi 22 , and thus also of Fi 23 and Fi 24 ′, and of the Baby Monster B . F 4 (2)′ is also a subquotient of the (pariah) Rudvalis group Ru , and has no involvements in sporadic simple groups except the ones already mentioned. The six exceptions are J 1 , J 3 , J 4 , O'N , Ru , and Ly , sometimes known as

874-523: The classification of finite simple groups is the Tits group T , which is sometimes considered of Lie type or sporadic — it is almost but not strictly a group of Lie type — which is why in some sources the number of sporadic groups is given as 27, instead of 26. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type, or both. The Tits group is the ( n = 0)-member F 4 (2)′ of

912-475: The cube of a purely imaginary number is also purely imaginary. For example, i = − i . The derivative of x equals 3 x . Cubes occasionally have the surjective property in other fields , such as in F p for such prime p that p ≠ 1 (mod 3) , but not necessarily: see the counterexample with rationals above . Also in F 7 only three elements 0, ±1 are perfect cubes, of seven total. −1, 0, and 1 are perfect cubes anywhere and

950-423: The cubes of consecutive integers can be expressed as follows: or There is no minimum perfect cube, since the cube of a negative integer is negative. For example, (−4) × (−4) × (−4) = −64 . Unlike perfect squares , perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25 , 75 and 00 can be the last two digits, any pair of digits with

988-403: The first n cubes is the n th triangle number squared: Proofs. Charles Wheatstone  ( 1854 ) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. He begins by giving the identity That identity is related to triangular numbers T n {\displaystyle T_{n}} in the following way: and thus

1026-434: The first one is a cube ( 1 = 1 ); the sum of the next two is the next cube ( 3 + 5 = 2 ); the sum of the next three is the next cube ( 7 + 9 + 11 = 3 ); and so forth. Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes: Every positive rational number

1064-498: The infinite family of commutator groups F 4 (2 )′ ; thus in a strict sense not sporadic, nor of Lie type. For n > 0 these finite simple groups coincide with the groups of Lie type F 4 (2 ), also known as Ree groups of type F 4 . The earliest use of the term sporadic group may be Burnside (1911 , p. 504) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay

1102-440: The last digit odd can occur as the last digits of a perfect cube. With even cubes, there is considerable restriction, for only 00 , o 2 , e 4 , o 6 and e 8 can be the last two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number (8 × 8) and a cube number (4 × 4 × 4) . This happens if and only if

1140-476: The last three digits of the number match 000, 043, 130, 213, 300, 343, 430, or 513. In decimal representation, 27 is located at the twenty-eighth (and twenty-ninth) digit after the decimal point in π : 3.141 592 653 589 793 238 462 643 383 27 9 … {\displaystyle 3.141\;592\;653\;589\;793\;238\;462\;643\;383\;{\color {red}27}9\ldots } If one starts counting with zero, 27

1178-539: The more recent mathematical literature, Stein (1971) uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also Benjamin, Quinn & Wurtz 2006 ); he observes that it may also be proved easily (but uninformatively) by induction, and states that Toeplitz (1963) provides "an interesting old Arabic proof". Kanim (2004) provides a purely visual proof, Benjamin & Orrison (2002) provide two additional proofs, and Nelsen (1993) gives seven geometric proofs. For example,

27 (number) - Misplaced Pages Continue

1216-550: The non-primitive ones can be trivially deduced from solutions for a smaller value of n . For example, for n = 24 , the solution 2 3 + 2 3 + 2 3 = 24 {\displaystyle 2^{3}+2^{3}+2^{3}=24} results from the solution 1 3 + 1 3 + 1 3 = 3 {\displaystyle 1^{3}+1^{3}+1^{3}=3} by multiplying everything by 8 = 2 3 . {\displaystyle 8=2^{3}.} Therefore, this

1254-417: The number is a perfect sixth power (in this case 2 ). The last digits of each 3rd power are: It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1 , 8 or 9 . That is their values modulo 9 may be only 0, 1, and 8. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3: It

1292-518: The only elements of a field equal to their own cubes: x − x = x ( x − 1)( x + 1) . Determination of the cubes of large numbers was very common in many ancient civilizations . Mesopotamian mathematicians created cuneiform tablets with tables for calculating cubes and cube roots by the Old Babylonian period (20th to 16th centuries BC). Cubic equations were known to the ancient Greek mathematician Diophantus . Hero of Alexandria devised

1330-403: The sporadic groups. The Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type , in which case there would be 27 sporadic groups. The monster group , or friendly giant , is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it. Five of the sporadic groups were discovered by Émile Mathieu in

1368-486: The sum of the first 5 cubes is the square of the 5th triangular number, A similar result can be given for the sum of the first y odd cubes, but x , y must satisfy the negative Pell equation x − 2 y = −1 . For example, for y = 5 and 29 , then, and so on. Also, every even perfect number , except the lowest, is the sum of the first 2 odd cubes ( p = 3, 5, 7, ...): There are examples of cubes of numbers in arithmetic progression whose sum

1406-482: The sum of three cubes. The smallest such integer for which such a sum is not known is 114. In September 2019, the previous smallest such integer with no known 3-cube sum, 42, was found to satisfy this equation: One solution to x 3 + y 3 + z 3 = n {\displaystyle x^{3}+y^{3}+z^{3}=n} is given in the table below for n ≤ 78 , and n not congruent to 4 or 5 modulo 9 . The selected solution

1444-410: The summands forming n 3 {\displaystyle n^{3}} start off just after those forming all previous values 1 3 {\displaystyle 1^{3}} up to ( n − 1 ) 3 {\displaystyle (n-1)^{3}} . Applying this property, along with another well-known identity: we obtain the following derivation: In

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