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From Wikipedia, the free encyclopedia

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

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Transcription

Contents

In mathematics

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 22 × (222 + 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.
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 → 62 + 82 = 100 → 12 + 02 + 02 = 1.

Other uses

See also

References

  1. ^ Sloane, N. J. A. (ed.). "Sequence 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, N. J. A. (ed.). "Sequence 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, N. J. A. (ed.). "Sequence A003401 (Numbers of edges of polygons constructible with ruler and compass)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A006355 (Number of binary vectors of length n containing no singletons)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ a b Sloane, N. J. A. (ed.). "Sequence A000260 (Number of rooted simplicial 3-polytopes with n+3 nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A004137 (Maximal number of edges in a graceful graph on n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A000664 (Number of graphs with n edges)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. ^ Sloane, N. J. A. (ed.). "Sequence 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, N. J. A. (ed.). "Sequence A007721 (Number of distinct degree sequences among all connected graphs with n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  11. ^ Sloane, N. J. A. (ed.). "Sequence A058673 (Number of matroids on n labeled points)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  12. ^ Sloane, N. J. A. (ed.). "Sequence 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 nth prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  13. ^ Sloane, N. J. A. (ed.). "Sequence 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, N. J. A. (ed.). "Sequence 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, N. J. A. (ed.). "Sequence 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.
This page was last edited on 19 August 2019, at 13:57
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