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6 (six) is the natural number following 5 and preceding 7. The SI prefix for 10006 is exa (E), and for its reciprocal atto- (a). ==In mathematics== 6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number; its proper divisors are 1, 2 and 3. Since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and -perfect number. As a perfect number: *6 is related to the Mersenne prime 3, since 21(22 - 1) = 6. (The next perfect number is 28.) *6 is the only even perfect number that is not the sum of successive odd cubes.〔David Wells, ''The Penguin Dictionary of Curious and Interesting Numbers''. London: Penguin Books (1987): 67〕 *As a perfect number, 6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one number; the square number, 25. Six is the only number that is both the sum and the product of three consecutive positive numbers.〔Peter Higgins, ''Number Story''. London: Copernicus Books (2008): 12〕 Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler."〔Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 72〕 Six is a congruent number. Six is the first discrete biprime (2.3) and the first member of the (2.q) discrete biprime family. Six is the smallest natural number that can be written as the sum of two positive rational cubes which are not integers: . Others up to 100 are: 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98. Six is a unitary perfect number, a harmonic divisor number and a superior highly composite number, the last to also be a primorial. The next superior highly composite number is 12. The next primorial is 30. 5 and 6 form a Ruth-Aaron pair under either definition. There are no Graeco-Latin squares with order 6. If ''n'' is a natural number that is not 2 or 6, than there is a Graeco-Latin square with order ''n''. The smallest non-abelian group is the symmetric group ''S''3 which has 3! = 6 elements. ''S''6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number ''n'' for which there is a construction of ''n'' isomorphic objects on an ''n''-set ''A'', invariant under all permutations of ''A'', but not naturally in 1-1 correspondence with the elements of ''A''. This can also be expressed category theoretically: consider the category whose objects are the ''n'' element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for ''n''=6. 6 similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others. 6 is the largest of the four all-Harshad numbers. A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of a power of 2 (namely 21) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon. Six is also an octahedral number. It is a triangular number and so is its square (36). There are six basic trigonometric functions. There are six convex regular polytopes in four dimensions. Six is the four-bit binary complement of number nine (6 = 01102 and 9 = 10012). The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials. All primes above 3 are of the form 6n±1 for n≥1. 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「6 (number)」の詳細全文を読む スポンサード リンク
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