216 (number)

216 (two hundred [and] sixteen) is the natural number following 215 and preceding 217. It is a cube, and is often called Plato's number, although it is not certain that this is the number intended by Plato.

In mathematics

Visual proof that 3³ + 4³ + 5³ = 6³

216 is the cube of 6, and the sum of three cubes:[math]\displaystyle{ 216=6^3=3^3+4^3+5^3. }[/math] It is the smallest cube that can be represented as a sum of three positive cubes,^[1] making it the first nontrivial example for Euler's sum of powers conjecture. It is, moreover, the smallest number that can be represented as a sum of any number of distinct positive cubes in more than one way.^[2] It is a highly powerful number: the product [math]\displaystyle{ 3\times 3 }[/math] of the exponents in its prime factorization [math]\displaystyle{ 216 = 2^3\times 3^3 }[/math] is larger than the product of exponents of any smaller number.^[3]

Because there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number.^[4] Although it is not a semiprime, the three closest numbers on either side of it are, making it the middle number between twin semiprime-triples, the smallest number with this property.^[5] Sun Zhiwei has conjectured that each natural number not equal to 216 can be written as either a triangular number or as a triangular number plus a prime number; however, this is not possible for 216. If the conjecture is true, 216 would be the only number for which this is not possible.^[6]

There are 216 ordered pairs of four-element permutations whose products generate all the other permutations on four elements.^[7] There are also 216 fixed hexominoes, the polyominoes made from 6 squares, joined edge-to-edge. Here "fixed" means that rotations or mirror reflections of hexominoes are considered to be distinct shapes.^[8]

In other fields

216 is one common interpretation of Plato's number, a number described in vague terms by Plato in the Republic. Other interpretations include 3600 and 12960000.^[9]

There are 216 colors in the web-safe color palette, a [math]\displaystyle{ 6\times 6\times 6 }[/math] color cube.^[10]

In the game of checkers, there are 216 different positions that can be reached by the first three moves.^[11]

The proto-Kabbalistic work Sefer Yetzirah states that the creation of the world was achieved by the manipulation of 216 sacred letters.^[12]

See also

• The year 216
• List of highways numbered 216
• All pages with titles containing 216

References

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/216 (number). Read more