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.

← 215 216 217 →
← 210 211 212 213 214 215 216 217 218 219 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	two hundred sixteen
Ordinal	216th (two hundred sixteenth)
Factorization	23 × 33
Divisors	1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
Greek numeral	ΣΙϚ´
Roman numeral	CCXVI, ccxvi
Binary	110110002
Ternary	220003
Senary	10006
Octal	3308
Duodecimal	16012
Hexadecimal	D816

In mathematics

Visual proof that 3³ + 4³ + 5³ = 6³

216 is the cube of 6, and the sum of three cubes:$${\displaystyle 216=6^{3}=3^{3}+4^{3}+5^{3}.}$$ It is the smallest cube that can be represented as a sum of three positive cubes,^[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 ${\displaystyle 3\times 3}$ of the exponents in its prime factorization ${\displaystyle 216=2^{3}\times 3^{3}}$ 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 ${\displaystyle 6\times 6\times 6}$ color cube.^[10]

See also

• All pages with titles containing 216