68 (sixty-eight) is the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

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 a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

In mathematics

68 is a Perrin number. 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 hold ...

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 direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

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 ideali ...

There are exactly 68 10-bit

binary number A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" ( zero) and "1" (one). The base-2 numeral system is a positional notati ...

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 In graph theory, a graceful labeling of a graph with edges is a labeling of its vertices with some subset of the integers from 0 to inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute differ ...

on 14 nodes has exactly 68 edges. There are 68 different

undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' ve ...

s with six edges and no isolated nodes, 68 different minimally 2-connected graphs on seven unlabeled nodes, 68 different

degree sequence In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is deno ...

s of four-node connected graphs, and 68 matroids on four labeled elements.

Størmer's theorem In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equ ...

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 three

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

's moves away from any cell. 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 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 number because ...

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

Other uses

*68 is the

atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of ever ...

erbium Erbium is a chemical element with the symbol Er and atomic number 68. A silvery-white solid metal when artificially isolated, natural erbium is always found in chemical combination with other elements. It is a lanthanide, a rare-earth element, ...

, 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". *68 may also be used as

slang Slang is vocabulary (words, phrases, and linguistic usages) of an informal register, common in spoken conversation but avoided in formal writing. It also sometimes refers to the language generally exclusive to the members of particular in-gr ...

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) and the throat. Cunnilingus is oral sex p ...

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

NCAA Division I men's basketball tournament The NCAA Division I men's basketball tournament, branded as NCAA March Madness and commonly called March Madness, is a single-elimination tournament played each spring in the United States, currently featuring 68 college basketball teams fro ...

has involved 68 teams in each edition since 2011, when the First Four round was introduced.

References

{{Integers, zero Integers

In mathematics

Other uses

See also

References