216 (number)

In mathematics

216 is the cube of 6, and the sum of three cubes:216=6^3=3^3+4^3+5^3. It is the smallest cube that can be represented as a sum of three positive cubes, 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. It is a highly powerful number: the product 3\times 3 of the exponents in its prime factorization 216 = 2^3\times 3^3 is larger than the product of exponents of any smaller number.

Because there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number. 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. 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.{{cite journal

There are 216 ordered pairs of four-element permutations whose products generate all the other permutations on four elements. 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.

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 and .{{cite journal

There are 216 colors in the web-safe color palette, a 6\times 6\times 6 color cube.{{cite journal

The international calling code for Tunisia

References

References

1. {{cite OEIS. A066890. Cubes that are the sum of three distinct positive cubes
2. {{cite OEIS. A003998. Numbers that are a sum of distinct positive cubes in more than one way
3. {{cite OEIS. A005934. Highly powerful numbers
4. {{cite OEIS. A005114. Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function
5. {{cite OEIS. A202319. Lesser of two semiprimes sandwiched each between semiprimes thus forming a twin semiprime-triple
6. {{cite OEIS. A071605. Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n
7. {{cite OEIS. A001168. Number of fixed polyominoes with n cells