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.
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 of the exponents in its
prime factorization 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]
^Sun, Zhi-Wei (2009). "On sums of primes and triangular numbers". Journal of Combinatorics and Number Theory. 1 (1): 65–76.
arXiv:0803.3737.
MR2681507.