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 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 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.