68 (number)

From Wikipedia, the free encyclopedia
Jump to: navigation, search
67 68 69
Cardinal sixty-eight
Ordinal 68th
(sixty-eighth)
Factorization 22× 17
Divisors 1, 2, 4, 17, 34, 68
Roman numeral LXVIII
Binary 10001002
Ternary 21123
Quaternary 10104
Quinary 2335
Senary 1526
Octal 1048
Duodecimal 5812
Hexadecimal 4416
Vigesimal 3820
Base 36 1W36

68 (sixty-eight) is the natural number following 67 and preceding 69.

In mathematics[edit]

68 is a Perrin number.[1]

It is the largest known number to be the sum of two primes in exactly two different ways: 68 = 7 + 61 = 31 + 37.[2] All higher even numbers that have been checked are the sum of three or more pairs of primes; the conjecture that 68 is the largest number with this property is closely related to the Goldbach conjecture and like it remains unproven.[3]

Because of the factorization of 68 as 2^2\cdot (2^{2^2}+1), a 68-sided regular polygon may be constructed with compass and straightedge.[4]

A Tamari lattice, with 68 upward paths of length zero or more from one element of the lattice to another.

There are exactly 68 10-bit binary numbers in which each bit has an adjacent bit with the same value,[5] exactly 68 combinatorially distinct triangulations of a given triangle with four points interior to it,[6]and exactly 68 intervals in the Tamari lattice describing the ways of parenthesizing five items.[6] The largest graceful graph on 13 nodes has exactly 68 edges.[7] There are 68 different undirected graphs with six edges and no isolated nodes,[8] 68 different minimally 2-connected graphs on seven unlabeled nodes,[9] 68 different degree sequences of four-node connected graphs,[10] and 68 matroids on four labeled elements.[11]

Størmer's theorem proves that, for every number p, there are a finite number of pairs of consecutive numbers that are both p-smooth (having no prime factor larger than p). For p = 13 this finite number is exactly 68.[12] On an infinite chessboard, there are 68 squares three knight's moves away from any cell.[13]

As a decimal number, 68 is the last two-digit number to appear in the digits of pi.[14] It is a happy number, meaning that repeatedly summing the squares of its digits eventually leads to 1:[15]

68\to 6^2+8^2=100\to 1^2+0^2+0^2=1.

Other uses[edit]

  • 68 is the atomic number of erbium, a lanthanide
  • In the restaurant industry, 68 may be used as a code meaning "put back on the menu", being the opposite of 86 which means "remove from the menu".[16]
  • 68 may also be used as slang for oral sex, based on a play on words involving the number 69.[17]

See also[edit]

References[edit]

  1. ^ "Sloane's A001608 : Perrin sequence (or Ondrej Such sequence): a(n) = a(n-2) + a(n-3)", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ http://math.fau.edu/richman/Interesting/WebSite/Number68.pdf retrieved 13 March 2013
  3. ^ "Sloane's A000954 : Conjecturally largest even integer which is an unordered sum of two primes in exactly n ways", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4. ^ "Sloane's A003401 : Numbers of edges of polygons constructible with ruler and compass", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ "Sloane's A006355 : Number of binary vectors of length n containing no singletons", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ a b "Sloane's A000260 : Number of rooted simplicial 3-polytopes with n+3 nodes", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. ^ "Sloane's A004137 : Maximal number of edges in a graceful graph on n nodes", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. ^ "Sloane's A000664 : Number of graphs with n edges", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. ^ "Sloane's A003317 : Number of unlabeled minimally 2-connected graphs with n nodes (also called "blocks")", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  10. ^ "Sloane's A007721 : Number of distinct degree sequences among all connected graphs with n nodes", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  11. ^ "Sloane's A058673 : Number of matroids on n labeled points", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  12. ^ "Sloane's A002071 : Number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most the n-th prime", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  13. ^ "Sloane's A018842 : Number of squares on infinite chess-board at n knight's moves from center", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  14. ^ "Sloane's A032510 : Scan decimal expansion of Pi until all n-digit strings have been seen; a(n) is last string seen", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  15. ^ "Sloane's A007770 : Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map includes 1", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  16. ^ Harrison, Mim (2009), Words at Work: An Insider’s Guide to the Language of Professions, Bloomsbury Publishing USA, p. 7, ISBN 9780802718686 .
  17. ^ Victor, Terry; Dalzell, Tom (2007), The Concise New Partridge Dictionary of Slang and Unconventional English (8th ed.), Psychology Press, p. 585, ISBN 9780203962114 .