68 (number)
| ||||
|---|---|---|---|---|
| Cardinal | sixty-eight | |||
| Ordinal |
68th (sixty-eighth) | |||
| Factorization | 22× 17 | |||
| Divisors | 1, 2, 4, 17, 34, 68 | |||
| Roman numeral | LXVIII | |||
| Binary | 10001002 | |||
| Ternary | 21123 | |||
| Quaternary | 10104 | |||
| Quinary | 2335 | |||
| Senary | 1526 | |||
| Octal | 1048 | |||
| Duodecimal | 5812 | |||
| Hexadecimal | 4416 | |||
| Vigesimal | 3820 | |||
| Base 36 | 1W36 | |||
68 (sixty-eight) is the natural number following 67 and preceding 69. It is an even number
In mathematics
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 22 × (222 + 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 → 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]
See also
References
- ↑ "Sloane's 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's 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's A003401 : Numbers of edges of polygons constructible with ruler and compass", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ "Sloane's A006355 : Number of binary vectors of length n containing no singletons", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 "Sloane's A000260 : Number of rooted simplicial 3-polytopes with n+3 nodes", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ "Sloane's A004137 : Maximal number of edges in a graceful graph on n nodes", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ "Sloane's A000664 : Number of graphs with n edges", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ "Sloane's A003317 : Number of unlabeled minimally 2-connected graphs with n nodes (also called "blocks")", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ "Sloane's A007721 : Number of distinct degree sequences among all connected graphs with n nodes", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ "Sloane's A058673 : Number of matroids on n labeled points", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ "Sloane's 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.
- ↑ "Sloane's 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's 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's 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.