216 (number)

From HandWiki
Short description: Natural number
← 215 216 217 →
Cardinaltwo hundred sixteen
Ordinal216th
(two hundred sixteenth)
Factorization23 × 33
Divisors1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
Greek numeralΣΙϚ´
Roman numeralCCXVI
Binary110110002
Ternary220003
Quaternary31204
Quinary13315
Senary10006
Octal3308
Duodecimal16012
HexadecimalD816
VigesimalAG20
Base 366036

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]

See also

References

  1. Sloane, N. J. A., ed. "Sequence A066890 (Cubes that are the sum of three distinct positive cubes)". OEIS Foundation. https://oeis.org/A066890. 
  2. Sloane, N. J. A., ed. "Sequence A003998 (Numbers that are a sum of distinct positive cubes in more than one way)". OEIS Foundation. https://oeis.org/A003998. 
  3. Sloane, N. J. A., ed. "Sequence A005934 (Highly powerful numbers)". OEIS Foundation. https://oeis.org/A005934. 
  4. Sloane, N. J. A., ed. "Sequence A005114 (Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function)". OEIS Foundation. https://oeis.org/A005114. 
  5. Sloane, N. J. A., ed. "Sequence A202319 (Lesser of two semiprimes sandwiched each between semiprimes thus forming a twin semiprime-triple)". OEIS Foundation. https://oeis.org/A202319. 
  6. "On sums of primes and triangular numbers". Journal of Combinatorics and Number Theory 1 (1): 65–76. 2009. 
  7. Sloane, N. J. A., ed. "Sequence A071605 (Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n)". OEIS Foundation. https://oeis.org/A071605. 
  8. Sloane, N. J. A., ed. "Sequence A001168 (Number of fixed polyominoes with n cells)". OEIS Foundation. https://oeis.org/A001168. 
  9. Adam, J. (February 1902). "The arithmetical solution of Plato's number". The Classical Review 16 (1): 17–23. 
  10. Thomas, B. (1998). "Palette's plunder". IEEE Internet Computing 2 (2): 87–89. doi:10.1109/4236.670691. 
  11. Sloane, N. J. A., ed. "Sequence A133047 (Starting from the standard 12 against 12 starting position in checkers, the sequence gives the number of distinct positions that can arise after n moves)". OEIS Foundation. https://oeis.org/A133047.