Number 6 in Math

A page of the Numeropedia - the Special Encyclopedia of Numbers

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3 factorial = 3! = 1×2×3 = 6 = 1+2+3.

 

A triangular number (sum of all integers from 1 to 3): 6 = 1+2+3 = 1×2×3. The pair 15 and 21 is the smallest pair of triangular numbers whose sum (36) and difference (6) are also triangular.

 

Product of the first 2 prime numbers: 6 = 2×3. The smallest composite number having at least 2 distinct prime factors.

 

The area of a right triangle of the smallest Pythagorean triple (3, 4, 5).

 

The smallest number whose cube is the sum of 3 consecutive cube numbers: 63 = 216 = 33+43+53.

 

The number of (square) faces on a cube (having 8 vertices/12 edges/6 faces).

 

The number of edges of a pyramid (or tetrahedron, 4 vertices/6 edges/4 equilateral trianglefaces).

 

The first perfect number: 6 = 1+2+3, i.e., a number equal to the sum of all of its proper divisors, including 1 (or also called aliquot divisors). Every even number of the form 2n–1(2n–1) is a perfect number if and only if (2n–1) is a prime number. It must be a triangular number: sum of all integers from 1 to (2n–1) and its last digit is either 6 or 8.  The first perfect numbers are: 6, 28, 496, 8128, 212(213–1) = 33550336, 216(217–1) = 8,589,869,056 and 218(219–1) = 137,438,691,328…

 

The number of vertices of the star (or shield) of David, a regular 6-pointed star polygon or hexagram. It is a sacred symbol in Judaism and Islam and is shown on the national flag of Israel.

 

Each ratio 17658/2943 = 27918/4653 = 34182/5697 = 6, using each of 9 digits 1-9 once, shows how to arrange a 9-book set on 2 shelves to mark the book #6.

 

A 6-pseudoparasite-triple number: 142,857×6 = 857,142.

 

688 = 8×86.

 

An automorphic number: its all powers end with 6.

 

64 = 1,296 = (1+2+9–6)4.

 

Solutions of the factorial equations n! = ab!: 6! = 5!×3! and 10! = 7!×6!.

 

The denominator of many Bernoulli numbers: B2 = 1/6, B14 = 7/6, B26 = 8553103/6, B34 = 2577687858367/6 and of B38, B62

 

195,287,346 = 6×32,547,891, using each of 9 digits 1-9 once on both sides of the equality.

 

85,427,136 = 2×42,713,568 = 4×21,356,784 = 6×14,237,856, using 3 permutations of the number itself (no digit 9 in all numbers).

 

6×7×9×524×831 = 164,597,832, using each of 9 digits 1-9 once on both sides of the equality.

 

The equality 5817×6 = 34,902 uses each of 10 digits 0-9 once.

 

The infinite sum Sn(1/n2) = p2/6 = z(2), where Sn is the infinite sum which n runs over all positive integers and z(s) is the Riemann zeta function over a complex variable s.

 

There are infinitely many square numbers that are the sum of squares of 2 consecutive numbers. Each root number in the series is 6 times the previous root number minus the earlier root number: 1, 52, 292, 1692

 

There are infinitely many square numbers that are also triangular numbers. The nth root number is 6 times the (n–1)th root minus the (n–2)th root: 1, 62, 352, 2042, 11892

 

The 2nd square number that is also a triangular number: 62 = 1+2+…+8 = 36.The nth root number is 6 times the (n–1)th root minus the (n–2)th root: 1, 62, 352, 2042, 11892

 

62 = 36 and if all digits are decremented (except the exponent) by 1: 52 = 25.

 

Sexy prime numbers are two prime numbers whose difference is 6.

 

Six circles theorem: Given a triangle and a circle tangent its two sides. Draw a circle tangent to the first circle and two other sides. Continue this way to form a chain, which ends with the sixth circle tangent the first circle.

 

The Lemoin circle goes through 6 points which formed by 3 lines going through the Lemoine point (or Grebe point, symmedian point) of a triangle and parallel to one side, intersecting two other sides.

 

The smallest case of 4 consecutive prime numbers in an arithmetic progression (of common difference 6): 251, 257, 263 and 269.

 

The smallest case of 3 consecutive prime numbers in an arithmetic progression (of common difference 6): 47, 53 and 59.

 

6.3..., 66.33..., 666.333..., ...

The square root of the number formed by n digits 4 followed by n digits 0 starts with n digits 6, decimal point, and n digits 3:

401/2 = 6.3…,

44001/2 = 66.33…,

4440001/2 = 666.333…,

444400001/2 = 6666.3333… 

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