# 68 (number)

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

| ||||
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Cardinal | sixty-eight | |||

Ordinal | 68th (sixty-eighth) | |||

Factorization | 2^{2} × 17 | |||

Divisors | 1, 2, 4, 17, 34, 68 | |||

Greek numeral | ΞΗ´ | |||

Roman numeral | LXVIII | |||

Binary | 1000100_{2} | |||

Ternary | 2112_{3} | |||

Quaternary | 1010_{4} | |||

Quinary | 233_{5} | |||

Senary | 152_{6} | |||

Octal | 104_{8} | |||

Duodecimal | 58_{12} | |||

Hexadecimal | 44_{16} | |||

Vigesimal | 38_{20} | |||

Base 36 | 1W_{36} |

## In mathematicsEdit

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 2^{2} × (2^{22} + 1), a 68-sided regular polygon may be constructed with compass and straightedge.^{[4]}

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 → 6
^{2}+ 8^{2}= 100 → 1^{2}+ 0^{2}+ 0^{2}= 1.

## Other usesEdit

- 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.

## See alsoEdit

## ReferencesEdit

**^**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.**^**http://math.fau.edu/richman/Interesting/WebSite/Number68.pdf retrieved 13 March 2013**^**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.**^**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.**^**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.- ^
^{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. **^**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.**^**Sloane, N. J. A. (ed.). "Sequence A000664 (Number of graphs with n edges)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**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.**^**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.**^**Sloane, N. J. A. (ed.). "Sequence A058673 (Number of matroids on n labeled points)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**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*n*th prime)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**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.**^**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.**^**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.**^**Harrison, Mim (2009),*Words at Work: An Insider’s Guide to the Language of Professions*, Bloomsbury Publishing USA, p. 7, ISBN 9780802718686.**^**Victor, Terry; Dalzell, Tom (2007),*The Concise New Partridge Dictionary of Slang and Unconventional English*(8th ed.), Psychology Press, p. 585, ISBN 9780203962114.