# 68 (number)

 ← 67 68 69 →
Cardinalsixty-eight
Ordinal68th
(sixty-eighth)
Factorization22 × 17
Divisors1, 2, 4, 17, 34, 68
Greek numeralΞΗ´
Roman numeralLXVIII
Binary10001002
Ternary21123
Quaternary10104
Quinary2335
Senary1526
Octal1048
Duodecimal5812
Vigesimal3820
Base 361W36

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]

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 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 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.

## References

1. ^ 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.
2. ^ http://math.fau.edu/richman/Interesting/WebSite/Number68.pdf retrieved 13 March 2013
3. ^
4. ^
5. ^
6. ^ a b
7. ^
8. ^ Sloane, N. J. A. (ed.). "Sequence A000664 (Number of graphs with n edges)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
9. ^
10. ^
11. ^ Sloane, N. J. A. (ed.). "Sequence A058673 (Number of matroids on n labeled points)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
12. ^
13. ^
14. ^
15. ^
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