6

Six is the smallest positive integer which is neither a square number nor a prime number. It is the second smallest composite number after four, equal to the sum and the product of its three proper divisors (1, 2 and 3).[1] As such, 6 is the only number that is both the sum and product of three consecutive positive numbers. It is the smallest perfect number, which are numbers that are equal to their aliquot sum, or sum of their proper divisors.[1][2] It is also the largest of the four all-Harshad numbers (1, 2, 4, and 6).[3]

6 is a pronic number and the only semiprime to be.[4] It is the first discrete biprime (2 × 3)[5] which makes it the first member of the (2 × q) discrete biprime family, where q is a higher prime. All primes above 3 are of the form 6n ± 1 for n ≥ 1.

As a perfect number:

• 6 is related to the Mersenne prime 3, since 2¹(2² – 1) = 6. (The next perfect number is 28.)
• 6 is the only even perfect number that is not the sum of successive odd cubes.[6]
• 6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, 25.

Six is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist; sixty (10 × 6) and ninety (15 × 6) are the next two.[7]

All integers ${\displaystyle n}$ that are multiples of 6 are pseudoperfect (all multiples of a pseudoperfect number are pseudoperfect). Six is also the smallest Granville number, or ${\displaystyle {\mathcal {S}}}$-perfect number.[8]

Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler".[9] Six is a congruent number.[10]

6 is the second primary pseudoperfect number,[11] and harmonic divisor number.[12] It is also the second superior highly composite number,[13] and the last to also be a primorial.

There are 6 non-equivalent ways in which 100 can be expressed as the sum of two prime numbers: (3 + 97), (11 + 89), (17 + 83), (29 + 71), (41 + 59) and (47 + 53).[14]

There is not a prime ${\displaystyle p}$ such that the multiplicative order of 2 modulo ${\displaystyle p}$ is 6, that is, ${\displaystyle ord_{p}(2)=6}$ By Zsigmondy's theorem, if ${\displaystyle n}$ is a natural number that is not 1 or 6, then there is a prime ${\displaystyle p}$ such that ${\displaystyle ord_{p}(2)=n}$. See A112927 for such ${\displaystyle p}$.

The ring of integer of the sixth cyclotomic field Q(ζ₆) , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω², where ${\displaystyle \omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}}$.

The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.[15]

There are six basic trigonometric functions: sin, cos, sec, csc, tan, and cot.[16]

The smallest non-abelian group is the symmetric group ${\displaystyle \mathrm {S_{3}} }$ which has 3! = 6 elements.[1]

Six is a triangular number[17] and so is its square (36). It is the first octahedral number, preceding 19.[18]

A regular cube, with six faces.

A six-sided polygon is a hexagon,[1] one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of a power of 2 (namely 2¹) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon with a compass and straightedge alone. A hexagram is a six-pointed geometric star figure (with the Schläfli symbol {6/2}, 2{3}, or {{3}}).

Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem.[19] The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.

There is only one non-trivial magic hexagon: it is of order-3 and made of nineteen cells, with a magic constant of 38. All rows and columns in a 6 × 6 magic square collectively generate a magic sum of 666 (which is doubly triangular). On the other hand, Graeco-Latin squares with order 6 do not exist; if ${\displaystyle n}$ is a natural number that is not 2 or 6, then there is a Graeco-Latin square of order ${\displaystyle n}$.[20]

The cube is one of five Platonic solids, with a total of six squares as faces. It is the only regular polyhedron that can generate a uniform honeycomb on its own, which is also self-dual. The cuboctahedron, which is an Archimedean solid that is one of two quasiregular polyhedra, has eight triangles and six squares as faces. Inside, its vertex arrangement can be interpreted as three hexagons that intersect to form an equatorial hexagonal hemi-face, by-which the cuboctahedron is dissected into triangular cupolas. This solid is also the only polyhedron with radial equilateral symmetry, where its edges and long radii are of equal length; its one of only four polytopes with this property — the others are the hexagon, the tesseract (as the four-dimensional analogue of the cube), and the 24-cell. Only six polygons are faces of non-prismatic uniform polyhedra such as the Platonic solids or the Archimedean solids: the triangle, the square, the pentagon, the hexagon, the octagon, and the decagon. If self-dual images of the tetrahedron are considered distinct, then there are a total of six regular polyhedra that are formed by three different Weyl groups in the third dimension (based on tetrahedral, octahedral and icosahedral symmetries).

How closely the shape of an object resembles that of a perfect sphere is called its sphericity, calculated by:[21]

$${\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}={\frac {A_{s}}{A_{p}}},}$$

where ${\displaystyle A_{s}}$ is the surface area of the sphere, ${\displaystyle V_{p}}$ the volume of the object, and ${\displaystyle A_{p}}$ the surface area of the object.

In four dimensions, there are a total of six convex regular polytopes: the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell.

${\displaystyle \mathrm {S_{6}} }$, with 720 = 6! elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4, the four-dimensional 5-cell, and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number ${\displaystyle n}$ for which there is a construction of ${\displaystyle n}$ isomorphic objects on an ${\displaystyle n}$-set ${\displaystyle A}$, invariant under all permutations of ${\displaystyle A}$, but not naturally in one-to-one correspondence with the elements of ${\displaystyle A}$. This can also be expressed category theoretically: consider the category whose objects are the ${\displaystyle n}$ element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for ${\displaystyle n=6}$.

In the classification of finite simple groups, twenty of twenty-six sporadic groups in the happy family are part of three families of groups which divide the order of the friendly giant, the largest sporadic group: five first generation Mathieu groups, seven second generation subquotients of the Leech lattice, and eight third generation subgroups of the friendly giant. The remaining six sporadic groups do not divide the order of the friendly giant, which are termed the pariahs (Ly, O'N, Ru, J₄, J₃, and J₁).[22]

The first appearance of 6 is in the Edicts of Ashoka c. 250 BCE. These are Brahmi numerals, ancestors of Hindu-Arabic numerals.

The first known digit "6" in the number "256" in Ashoka's Minor Rock Edict No.1 in Sasaram, c. 250 BCE

The evolution of our modern digit 6 appears rather simple when compared with the other digits. The modern 6 can be traced back to the Brahmi numerals of India, which are first known from the Edicts of Ashoka c. 250 BCE.[29][30][31][32] It was written in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.[33]

On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a "b" is not practical.

Just as in most modern typefaces, in typefaces with text figures the character for the digit 6 usually has an ascender, as, for example, in .[34]

This digit resembles an inverted 9. To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels.

• The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit.[72] They are the oldest remaining teams in the league, though not necessarily the first six; they comprised the entire league from 1942 to 1967.

• Number of players:
  • In association football (soccer), the number of substitutes combined by both teams, that are allowed in the game.
  • In box lacrosse, the number of players per team, including the goaltender, that are on the floor at any one time, excluding penalty situations.[73]
  • In ice hockey, the number of players per team, including the goaltender, that are on the ice at any one time during regulation play, excluding penalty situations. (Some leagues reduce the number of players on the ice during overtime.)[74]
  • In volleyball:
    • Six players from each team on each side play against each other.[75]
    • Standard rules only allow six total substitutions per team per set. (Substitutions involving the libero, a defensive specialist who can only play in the back row, are not counted against this limit.)
  • Six-man football is a variant of American or Canadian football, played by smaller schools with insufficient enrollment to field the traditional 11-man (American) or 12-man (Canadian) squad.[76]
• Scoring:
  • In both American and Canadian football, 6 points are awarded for a touchdown.[77]
  • In Australian rules football, six points are awarded for a goal, scored when a kicked ball passes between the defending team's two inner goalposts without having been touched by another player.
  • In cricket, six runs are scored for the batting team when the ball is hit to the boundary or the ground beyond it without having touched the ground in the field.
• In basketball, the ball used for women's full-court competitions is designated "size 6".[78]
• In pool and snooker one its table contains six pockets.
• In most rugby league competitions (but not the Super League, which uses static squad numbering), the jersey number 6 is worn by the starting five-eighth (Southern Hemisphere term) or stand-off (Northern Hemisphere term).
• In rugby union, the starting blindside flanker wears jersey number 6. (Some teams use "left" and "right" flankers instead of "openside" and "blindside", with 6 being worn by the starting left flanker.)[79]

6 as a resin identification code, used in recycling.

• On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry Pearl it is the key for J and K, and on the BlackBerry 8700 series and Curve 8900 with full keyboard, it is the key for F
• The "6-meter band" in amateur radio includes the frequencies from 50 to 54 MHz
• 6 is the resin identification code used in recycling to identify polystyrene[80]

• In the ancient Roman calendar, Sextilis was the sixth month. After the Julian reform, June became the sixth month and Sextilis was renamed August[81]
• Sextidi was the sixth day of the décade in the French Revolutionary calendar[82]

• The name of the smallest group of Cub Scouts and Guiding's equivalent Brownies, traditionally consisting of six people and is led by a "sixer".
• A coffin is traditionally buried six feet under the ground; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead[95]
• There are said to be no more than six degrees of separation between any two people on Earth.[96]
• In Western astrology, Virgo is the 6th astrological sign of the Zodiac[97]
• The Six Dynasties form part of Chinese history[98]
• Six is a lucky number in Chinese culture.[99]
• The Birmingham Six were a British miscarriage of justice, held in prison for 16 years.[100]
• "Six" is used as an informal slang term for the British Secret Intelligence Service, MI6.[101]

International maritime signal flag for 6

• Six pack is a common form of packaging for six bottles or cans of drink (especially beer), and by extension, other assemblages of six items.[102] Also, six is half a dozen.
• In Pythagorean numerology, the number 6 is the digit of balance, harmony and organization of the home and family
• The fundamental flight instruments lumped together on a cockpit display are often called the Basic Six or six-pack.
• The number of dots in a braille cell.[103]
  • See also Six degrees (disambiguation).
• Extrasensory perception is sometimes called the "sixth sense".[104]
• Six Flags is an American company running amusement parks and theme parks in the U.S., Canada, and Mexico.[105]
• In the U.S. Army "Six" as part of a radio call sign is used by the commanding officer of a unit, while subordinate platoon leaders usually go by "One".[106] (For a similar example see also: Rainbow Six.)

• List of highways numbered 6

• The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66–68
• A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links ISBN 0-521-42385-6
• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 67 - 69

Look up six in Wiktionary, the free dictionary.

• The Number 6
• The Positive Integer 6
• Prime curiosities: 6