68 is acomposite 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).
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 theGoldbach conjecture and, like it, remains unproven.[3]
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-bitbinary numbers in which each bit has an adjacent bit with the same value,[5] exactly 68 combinatorially distincttriangulations of a given triangle with four points interior to it,[6] and exactly 68intervals in theTamari lattice describing the ways of parenthesizing five items.[6] The largestgraceful graph on 14 nodes has exactly 68 edges.[7] There are 68 differentundirected graphs with six edges and no isolated nodes,[8] 68 different minimally2-connected graphs on seven unlabeled nodes,[9] 68 differentdegree sequences of four-node connected graphs,[10] and 68matroids on four labeled elements.[11]
Størmer's theorem proves that, for every numberp, there are a finite number of pairs of consecutive numbers that are bothp-smooth (having no prime factor larger thanp). Forp = 13 this finite number is exactly 68.[12] On an infinite chessboard, there are 68 squares that are threeknight's moves away from any starting square.[13]
As adecimal number, 68 is the last two-digit number to appear for the first time in the digits ofpi.[14] It is ahappy number, meaning that repeatedly summing the squares of its digits eventually leads to 1:[15]
