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24 (twenty-four) is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta (Y), and for 10−24 (i.e., the reciprocal of 1024) yocto (y). These numbers are the largest and smallest number to receive an SI prefix to date. ==In mathematics== *24 is the factorial of 4 (24 = 4!) and a composite number, being the first number of the form , where is an odd prime. *Since 24 = 4!, it follows that 24 is the number of ways to order 4 distinct items: (1,2,3,4), (1,2,4,3), (1,3,2,4,), (1,3,4,2), (1,4,2,3), (1,4,3,2), (2,1,3,4,), (2,1,4,3), (2,3,1,4,), (2,3,4,1), (2,4,1,3), (2,4,3,1), (3,1,2,4), (3,1,4,2), (3,2,1,4), (3,2,4,1), (3,4,1,2), (3,4,2,1), (4,1,2,3), (4,1,3,2), (4,2,1,3), (4,2,3,1), (4,3,1,2), (4,3,2,1). *It is the smallest number with exactly eight divisors: 1, 2, 3, 4, 6, 8, 12, and 24. *It is a highly composite number, having more divisors than any smaller number. *24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. *Subtracting 1 from any of its divisors (except 1 and 2, but including itself) yields a prime number; 24 is the largest number with this property. *24 has an aliquot sum of 36 and the aliquot sequence (24, 36, 55, 17, 1, 0). It is therefore the lowest abundant number whose aliquot sum is itself abundant. *The aliquot sum of only one number, 529 = 232, is 24. *There are 10 solutions to the equation φ(''x'') = 24, namely 35, 39, 45, 52, 56, 70, 72, 78, 84 and 90. This is more than any integer below 24, making 24 a highly totient number. *24 is a nonagonal number. *24 is the sum of the prime twins 11 and 13. *24 is a Harshad number. *24 is a semi-meandric number. *The product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two even numbers, one of which is a multiple of four, and there must be a multiple of three. *The tesseract has 24 two-dimensional faces (which are all squares). *24 is the only nontrivial solution to the cannonball problem, that is: 12+22+32+...+242 is a perfect square (=702). (The trivial case is just 12 = 12.) *In 24 dimensions there are 24 even positive definite unimodular lattices, called the Niemeier lattices. One of these is the exceptional Leech lattice which has many surprising properties; due to its existence, the answers to many problems such as the kissing number problem and densest lattice sphere-packing problem are known in 24 dimensions but not in many lower dimensions. The Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S(5,8,24) and the Mathieu group M24. (One construction of the Leech lattice is possible because 12+22+32+...+242 =702.) *The modular discriminant Δ(τ) is proportional to the 24th power of the Dedekind eta function η(τ): Δ(τ) = (2π)12η(τ)24. *The Barnes-Wall lattice contains 24 lattices. *24 is the only number whose divisors — namely, — are exactly those numbers ''n'' for which every invertible element of the commutative ring Z/''n''Z is a square root of 1. It follows that the multiplicative group (Z/24Z)× = is isomorphic to the additive group (Z/2Z)3. This fact plays a role in monstrous moonshine. *:It follows that any number ''n'' relatively prime to 24, and in particular any prime ''n'' greater than 3, has the property that ''n''2-1 is divisible by 24. *The 24-cell, with 24 octahedral cells and 24 vertices, is a self-dual convex regular 4-polytope; it has no analogue in any other dimension. Its 24 vertices can be expressed as the set of unit quaternions, using all choices of signs. This set forms a group under quaternion multiplication, isomorphic to the binary tetrahedral group. The quotient of the unit quaternions S3 by this subgroup is identical as a metric space to the configuration space of a regular tetrahedron centered at the origin in 3-space. The 24-cell tiles 4-dimensional space. *24 is the kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 24 such spheres form the vertices of a 24-cell.) *24 is the second Granville number, the previous being 6 and the next being 28. It is the first Granville number that is not also a conventional perfect number. *24 is the largest integer that is evenly divisible by all natural numbers no larger than its square root. *24 is the Euler characteristic of a K3 surface 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「24 (number)」の詳細全文を読む スポンサード リンク
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