6 (number) - in Mathematics

In Mathematics

6 is the first number 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.
• 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.

Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler." 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 a unitary perfect number, a harmonic divisor number and a highly composite number. The next highly composite number is 12.

5 and 6 form a Ruth-Aaron pair under either definition.

The smallest non-abelian group is the symmetric group S₃ which has 3! = 6 elements.

S₆, 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. 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:

The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.