68 (number)

Last updated
← 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
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

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 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

See also

Related Research Articles

10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.

222 is the natural number following 221 and preceding 223.

69 (sixty-nine) is the natural number following 68 and preceding 70.

64 (sixty-four) is the natural number following 63 and preceding 65.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

300 is the natural number following 299 and preceding 301.

700 is the natural number following 699 and preceding 701.

800 is the natural number following 799 and preceding 801.

185 is the natural number following 184 and preceding 186.

<span class="mw-page-title-main">1,000,000</span> Natural number

One million (1,000,000), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione, from mille, "thousand", plus the augmentative suffix -one.

<span class="mw-page-title-main">1,000,000,000</span> Natural number

1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. With a number, "billion" can be abbreviated as b, bil or bn.

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

10,000,000 is the natural number following 9,999,999 and preceding 10,000,001.

100,000,000 is the natural number following 99,999,999 and preceding 100,000,001.

177 is the natural number following 176 and preceding 178.

202 is the natural number following 201 and preceding 203.

209 is the natural number following 208 and preceding 210.

744 is the natural number following 743 and preceding 745.

60,000 (sixty thousand) is the natural number that comes after 59,999 and before 60,001. It is a round number. It is the value of (F25).

888 is the natural number following 887 and preceding 889.

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
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.