68 (number)

From HandWiki
Short description: Natural number
← 67 68 69 →
Cardinalsixty-eight
Ordinal68th
(sixty-eighth)
Factorization22 × 17
Divisors1, 2, 4, 17, 34, 68
Greek numeralΞΗ´
Roman numeralLXVIII
Binary10001002
Ternary21123
Quaternary10104
Quinary2335
Senary1526
Octal1048
Duodecimal5812
Hexadecimal4416
Vigesimal3820
Base 361W36

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

In mathematics

68 is a composite number; a square-prime, of the form (p2, q) where q is a higher prime. It is the eighth of this form and the sixth of the form (22.q).

68 is a Perrin number.[1]

It has an aliquot sum of 58 within an aliquot sequence of two composite numbers (68, 58,32,31,1,0) to the Prime in the 31-aliquot tree.

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

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 14 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 for the first time 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

  • 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]
  • The NCAA Division I men's basketball tournament has involved 68 teams in each edition since 2011, when the First Four round was introduced.
  • The NCAA Division I women's basketball tournament expanded to 68 teams in 2022, matching the men's tournament.

See also

  • 68 (disambiguation)

References

  1. ↑ Sloane, N. J. A., ed. "Sequence A001608 (Perrin sequence (or Ondrej Such sequence): a(n) = a(n-2) + a(n-3))". OEIS Foundation. https://oeis.org/A001608. 
  2. ↑ "68 Sixty-Eight LXVIII". http://math.fau.edu/richman/Interesting/WebSite/Number68.pdf. 
  3. ↑ Sloane, N. J. A., ed. "Sequence A000954 (Conjecturally largest even integer which is an unordered sum of two primes in exactly n ways)". OEIS Foundation. https://oeis.org/A000954. 
  4. ↑ Sloane, N. J. A., ed. "Sequence A003401 (Numbers of edges of polygons constructible with ruler and compass)". OEIS Foundation. https://oeis.org/A003401. 
  5. ↑ Sloane, N. J. A., ed. "Sequence A006355 (Number of binary vectors of length n containing no singletons)". OEIS Foundation. https://oeis.org/A006355. 
  6. ↑ 6.0 6.1 Sloane, N. J. A., ed. "Sequence A000260 (Number of rooted simplicial 3-polytopes with n+3 nodes)". OEIS Foundation. https://oeis.org/A000260. 
  7. ↑ Sloane, N. J. A., ed. "Sequence A004137 (Maximal number of edges in a graceful graph on n nodes)". OEIS Foundation. https://oeis.org/A004137. 
  8. ↑ Sloane, N. J. A., ed. "Sequence A000664 (Number of graphs with n edges)". OEIS Foundation. https://oeis.org/A000664. 
  9. ↑ Sloane, N. J. A., ed. "Sequence A003317 (Number of unlabeled minimally 2-connected graphs with n nodes (also called "blocks"))". OEIS Foundation. https://oeis.org/A003317. 
  10. ↑ Sloane, N. J. A., ed. "Sequence A007721 (Number of distinct degree sequences among all connected graphs with n nodes)". OEIS Foundation. https://oeis.org/A007721. 
  11. ↑ Sloane, N. J. A., ed. "Sequence A058673 (Number of matroids on n labeled points)". OEIS Foundation. https://oeis.org/A058673. 
  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)". OEIS Foundation. https://oeis.org/A002071. 
  13. ↑ Sloane, N. J. A., ed. "Sequence A018842 (Number of squares on infinite chess-board at n knight's moves from center)". OEIS Foundation. https://oeis.org/A018842. 
  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)". OEIS Foundation. https://oeis.org/A032510. 
  15. ↑ Sloane, N. J. A., ed. "Sequence A007770 (Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map includes 1)". OEIS Foundation. https://oeis.org/A007770. 
  16. ↑ Harrison, Mim (2009), Words at Work: An Insider's Guide to the Language of Professions, Bloomsbury Publishing USA, p. 7, ISBN 9780802718686, https://books.google.com/books?id=dAf_HxF1HTwC&pg=PA7 .
  17. ↑ Victor, Terry; Dalzell, Tom (2007), The Concise New Partridge Dictionary of Slang and Unconventional English (8th ed.), Psychology Press, p. 585, ISBN 9780203962114, https://books.google.com/books?id=7UIjVGcSe8MC&pg=PA585 



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