Английская Википедия:68 (number)
68 (sixty-eight) is the natural number following 67 and preceding 69. It is an even number.
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 Шаблон:Nowrap, 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 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 → 6Шаблон:Sup + 8Шаблон:Sup = 100 → 1Шаблон:Sup + 0Шаблон:Sup + 0Шаблон:Sup = 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]
- 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
References
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