68 (sixty-eight) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following 67 and preceding 69. It is an

even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

. 68 is a

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...

and the eighth square-prime.

In mathematics

68 is a

; a square-prime, of the form (''p''², 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. It has an

aliquot sum In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. That is, s(n)=\sum_ d \, . It can be used to characterize the prime numbers, perfect numbers, sociabl ...

of 58 within an

aliquot sequence In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Def ...

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. 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 Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to ho ...

and, like it, remains unproven. Because of the factorization of 68 as , a 68-sided

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

may be constructed with

compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...

There are exactly 68 10-bit

binary number A binary number is a number expressed in the Radix, base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may ...

s in which each bit has an adjacent bit with the same value, exactly 68 combinatorially distinct triangulations of a given triangle with four points interior to it, and exactly 68 intervals in the Tamari lattice describing the ways of parenthesizing five items. The largest graceful graph on 14 nodes has exactly 68 edges. There are 68 different

undirected graph In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also call ...

s with six edges and no isolated nodes, 68 different minimally 2-connected graphs on seven unlabeled nodes, 68 different degree sequences of four-node connected graphs, and 68

matroid In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most significant being in terms ...

s on four labeled elements. 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. On an infinite chessboard, there are 68 squares that are three

knight A knight is a person granted an honorary title of a knighthood by a head of state (including the pope) or representative for service to the monarch, the church, or the country, especially in a military capacity. The concept of a knighthood ...

's moves away from any starting square. As a decimal number, 68 is the last two-digit number to appear for the first time in the digits of pi. It is a

happy number In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy ...

, meaning that repeatedly summing the squares of its digits eventually leads to 1: :68 → 6 + 8 = 100 → 1 + 0 + 0 = 1.

Other uses

*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". *68 may also be used as

slang A slang is a vocabulary (words, phrases, and linguistic usages) of an informal register, common in everyday conversation but avoided in formal writing and speech. It also often refers to the language exclusively used by the members of pa ...

for

oral sex Oral sex, sometimes referred to as oral intercourse, is sexual activity involving the stimulation of the genitalia of a person by another person using the mouth (including the lips, tongue, or teeth). Cunnilingus is oral sex performed on the vu ...

, based on a play on words involving the number 69.

References

{{Integers, zero Integers

In mathematics

Other uses

See also

References