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68 (number)

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← 67 68 69 →
Cardinalsixty-eight
Ordinal68th
(sixty-eighth)
Factorization22 × 17
Divisors1, 2, 4, 17, 34,40
Greek numeralΞΗ´
Roman numeralLXVIII
Binary10001002
Ternary21123
Senary1526
Octal1048
Duodecimal5812
Hexadecimal4416

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

In mathematics

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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 that are three knight's moves away from any starting square.[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

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See also

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References

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  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. ^ "68 Sixty-Eight LXVIII" (PDF). math.fau.edu. 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
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