68 (number)
| ||||
|---|---|---|---|---|
| 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.
In mathematics
[edit]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]
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
[edit]- 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.
- The NCAA Division I women's basketball tournament expanded to 68 teams in 2022, matching the men's tournament.
See also
[edit]References
[edit]- ^ 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.
- ^ "68 Sixty-Eight LXVIII" (PDF). math.fau.edu. 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 nth 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