216 (two hundred ndsixteen) is the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

following 215 and preceding

217 Year 217 ( CCXVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Praesens and Extricatus (or, less frequently, year 970 ''Ab urbe ...

. It is a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

, and is often called

Plato's number Plato's number is a number enigmatically referred to by Plato in his dialogue the ''Republic'' (8.546b). The text is notoriously difficult to understand and its corresponding translations do not allow an unambiguous interpretation. There is no rea ...

, although it is not certain that this is the number intended by

Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...

In mathematics

216 is the

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 In elementary number theory, a highly powerful number is a positive integer that satisfies a property introduced by the Indo-Canadian mathematician Mathukumalli V. Subbarao. The set of highly powerful numbers is a proper subset of the set of power ...

: the product

3\times 3

of the exponents in its

prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are s ...

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 divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s of any other integer, it is an untouchable number. Although it is not a

semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...

, 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 A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...

or as a triangular number plus a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

; however, this is not possible for 216. If the conjecture is true, 216 would be the only number for which this is not possible. There are 216 ordered pairs of four-element

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...

s whose products generate all the other permutations on four elements. There are also 216 fixed

hexomino A hexomino (or 6-omino) is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are n ...

es, the

polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in pop ...

es 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

, a number described in vague terms by

in the ''

Republic A republic () is a " state in which power rests with the people or their representatives; specifically a state without a monarchy" and also a "government, or system of government, of such a state." Previously, especially in the 17th and 18th ...

''. Other interpretations include and . There are 216 colors in the web-safe color palette, a

6\times 6\times 6

color cube. In the game of

checkers Checkers (American English), also known as draughts (; British English), is a group of strategy board games for two players which involve diagonal moves of uniform game pieces and mandatory captures by jumping over opponent pieces. Checkers ...

, there are 216 different positions that can be reached by the first three moves.

References

{{DEFAULTSORT:216 (Number) Integers

In mathematics

In other fields

See also

References