68 (number)

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← 67 68 69 →
Cardinal sixty-eight
Ordinal 68th
(sixty-eighth)
Factorization 22 × 17
Divisors 1, 2, 4, 17, 34, 68
Greek numeral ΞΗ´
Roman numeral LXVIII, lxviii
Binary 10001002
Ternary 21123
Senary 1526
Octal 1048
Duodecimal 5812
Hexadecimal 4416

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

Contents

68 is a composite number and the eighth square-prime.

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 Tamari lattice.svg
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

See also

References

  1. ↑ Sloane, N. J. A. (ed.). "SequenceA001608(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.). "SequenceA000954(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.). "SequenceA003401(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.). "SequenceA006355(Number of binary vectors of length n containing no singletons)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  6. 1 2 Sloane, N. J. A. (ed.). "SequenceA000260(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.). "SequenceA004137(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.). "SequenceA000664(Number of graphs with n edges)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  9. ↑ Sloane, N. J. A. (ed.). "SequenceA003317(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.). "SequenceA007721(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.). "SequenceA058673(Number of matroids on n labeled points)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  12. ↑ Sloane, N. J. A. (ed.). "SequenceA002071(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.). "SequenceA018842(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.). "SequenceA032510(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.). "SequenceA007770(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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