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, six is the only number that is both the sum and product of three consecutive positive numbers. 6 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 the largest of the four all-Harshad numbers (1, 2, 4, and 6),^[3] where it represents the sum between the first prime and composite, 2 and 4.

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 (that is also the number of regions formed by the diagonals and edges of a regular six-sided hexagon).^[7]

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.^[8]

It is the first primitive pseudoperfect number,^[9] and all integers ${\displaystyle n}$  that are multiples of 6 are pseudoperfect (all multiples of a perfect number are pseudoperfect); six is also the smallest Granville number, or ${\displaystyle {\mathcal {S}}}$ -perfect number.^[10]

Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler".^[11] Six is a congruent number.^[12]

6 is the second primary pseudoperfect number,^[13] and harmonic divisor number.^[14] It is also the second superior highly composite number,^[15] 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).^[16]

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}  ByZsigmondy'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.^[17]

There are six basic trigonometric functions: sin, cos, sec, csc, tan, and cot.^[18]

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

Six is a triangular number^[19] and so is its square (36). It is the first octahedral number, preceding 19.^[20]

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.^[21] 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 sumof666 (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 ${\displaystyle n}$ .^[22]

The cube is one of five Platonic solids, with a total of six squaresasfaces. 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:^[23]

Infour 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).^[24]

Hexa is classical Greek for "six".^[1] Thus:

• "Hexadecimal" combines hexa- with the Latinate decimal to name a number base of 16^[25]
• Ahexagon is a regular polygon with six sides^[26]
  • L'Hexagone is a French nickname for the continental part of Metropolitan France for its resemblance to a regular hexagon
• Ahexahedron is a polyhedron with six faces, with a cube being a special case^[27]
• Hexameter is a poetic form consisting of six feet per line
• A "hex nut" is a nut with six sides, and a hex bolt has a six-sided head
• The prefix "hexa-" also occurs in the systematic name of many chemical compounds, such as hexane which has 6 carbon atoms (C₆H₁₄).

Sex- is a Latin prefix meaning "six".^[1] Thus:

• Senary is the ordinal adjective meaning "sixth"^[28]
• People with sexdactyly have six fingers on each hand
• The measuring instrument called a sextant got its name because its shape forms one-sixth of a whole circle
• A group of six musicians is called a sextet
• Six babies delivered in one birth are sextuplets
• Sexy prime pairs – Prime pairs differing by six are sexy, because sex is the Latin word for six.^[29][30]

The SI prefix for 1000⁶isexa- (E), and for its reciprocal atto- (a).

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 numeralsofIndia, which are first known from the Edicts of Ashoka c. 250 BCE.^{[31][32][33][34]} 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.^[35]

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  .^[36]

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.

• Les Six ("The Six" in English) was a group consisting of the French composers Georges Auric, Louis Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc and Germaine Tailleferre in the 1920s^[37]
• Bands with the number six in their name include Six Organs of Admittance,^[38] 6 O'Clock Saints, Electric Six,^[39] Eve 6, Los Xey (sei is Basque for "six"), Out On Blue Six, Six In Six, Sixpence None the Richer,^[40] Slant 6,^[41] Vanity 6, and You Me At Six^[42]
• #6 is the pseudonym of American musician Shawn Crahan, when performing with the band Slipknot

• A standard guitar has six strings^[43]
• Most woodwind instruments have six basic holes or keys (e.g., bassoon, clarinet, pennywhistle, saxophone); these holes or keys are usually not given numbers or letters in the fingering charts

• There are six whole tones in an octave.^[44]
• There are six semitones in a tritone.^[45]

• "Six geese a-laying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas".^[46]
• Divided in six arias, Hexachordum Apollinis is generally regarded as one of the pinnacles of Johann Pachelbel's oeuvre.^[47]
• The theme of the sixth album by Dream Theater, Six Degrees of Inner Turbulence, was the number six: the album has six songs, and the sixth song—that is, the complete second disc—explores the stories of six individuals suffering from various mental illnesses.^[48]
• Aristotle gave six elements of tragedy, the first of which is Mythos.^[49]

• Six points on a Star of David^[50]
• Six orders of the Mishnah^[51]
• Six symbolic foods placed on the Passover Seder Plate^[52]
• God took six days to create the world in the Old Testament Book of Genesis;^[53] humankind was created on day 6. In the City of God, Augustine of Hippo suggested (book 11, chapter 30) that God's creation of the world took six days because 6 is a perfect number.^[54]
• The Jewish holiday of Shavuot starts on the sixth day of the Hebrew month of Sivan^[55]
• Seraphs have six wings.^[56]

• There are Six articles of faith^[57]
• Six points on a Seal of Solomon
• Fasting six days of Shawwal, together with the month of Ramadan, is equivalent to fasting the whole year^[58]
• The heavens and earth were created in six days according to the Quran:^[59]

Indeed, We created the heavens and the earth and everything in between in six Days,¹ and We were not ˹even˺ touched with fatigue.²

— Surah Qaf:38

Note 1: The word day is not always used in the Quran to mean a 24-hour period. According to Surah Al-Hajj (The Pilgrimage):47, a heavenly Day is 1000 years of our time. The Day of Judgment will be 50,000 years of our time - Surah Al-Maarij (The Ascending Stairways):4. Hence, the six Days of creation refer to six eons of time, known only by Allah.

Note 2: Some Islamic scholars believe this verse comes in response to Exodus 31:17, which says, "The Lord made the heavens and the earth in six days, but on the seventh day He rested and was refreshed."

• InHindu theology, a trasarenu is the combination of six celestial paramānus (atoms).
• In Taoism:
  • Six Lines of a Hexagram
  • Six MinistriesofHuang Di^[60]

• Messier object M6, a magnitude 4.5 open cluster in the constellation Scorpius, also known as the Butterfly Cluster^[61]
• The New General Catalogue object NGC 6, a spiral galaxy in the constellation Andromeda
• The Roman numeral VI:
  • Stands for subdwarfs in the Yerkes spectral classification scheme
  • (Usually) stands for the sixth-discovered satellite of a planet or minor planet (e.g. Jupiter VI)
• 6 Hebe

• The cells of a beehive are six-sided.^[62]
• Insects have six legs (see Hexapoda).^[63]
• Six kingdoms in the taxonomic rank below domain (biology); Animalia, Plantae, Fungi, Protista, Archaea/Archaeabacteria, and Bacteria/Eubacteria. See Kingdom (biology).^[64]
• The six elements most common in biomolecules are called the CHNOPS elements; the letters stand for the chemical abbreviations of carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur. See CHON.^[65]

• A benzene molecule has a ring of six carbon atoms.^[66]
• 6 is the atomic number of carbon.^[67]
• The sixfold symmetryofsnowflakes arises from the hexagonal crystal structureofordinary ice.^[68]
• Ahexamer is an oligomer made of six subunits.

• There are six tastes in traditional Indian medicine (Ayurveda): sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body.^[69]
• Phase 6 is one of six pandemic influenza phases.^[70]

• In the Standard Model of particle physics, there are six types of quarks and six types of leptons.^[71]
• Instatistical mechanics, the six-vertex model has six possible configurations of arrows at each vertex^[72]
• There are six colors in the RGB color wheel: (primary) red, blue, green, (secondary) cyan, magenta, and yellow. (See Tertiary color)^[73]
• Inthree-dimensional Euclidean space, there are six unknown support reactions for a statically determinate structure: one force in each of the three dimensions, and one moment through each of three possible orthogonal planes.

• The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit.^[74] They are the oldest remaining teams in the league, though not necessarily the first six; they comprised the entire league from 1942to1967.
• Number of players:
  • Inassociation football (soccer), the number of substitutes combined by both teams, that are allowed in the game.
  • Inbox lacrosse, the number of players per team, including the goaltender, that are on the floor at any one time, excluding penalty situations.^[75]
  • Inice 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.)^[76]
  • Involleyball:
    • Six players from each team on each side play against each other.^[77]
    • 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.^[78]
• Scoring:
  • In both American and Canadian football, 6 points are awarded for a touchdown.^[79]
  • InAustralian 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.
  • Incricket, 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.
• Inbasketball, the ball used for women's full-court competitions is designated "size 6".^[80]
• Inpool 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).
• Inrugby 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.)^[81]

• 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^[82]

• In the ancient Roman calendar, Sextilis was the sixth month. After the Julian reform, June became the sixth month and Sextilis was renamed August^[83]
• Sextidi was the sixth day of the décade in the French Revolutionary calendar^[84]

• The number of sides on a cube, hence the highest number on a standard die^[85]
• The six-sided tiles on a hex grid are used in many tabletop and board games.
• The highest number on one end of a standard domino

• The Power of Six is a book written by Pittacus Lore, and the second in the Lorien Legacies series.^[86]
• Number 6 is a character in the book series Lorien Legacies.

• Number Six (Tricia Helfer) is a family of fictional characters from the reimagined science fiction television series, Battlestar Galactica.
• Number 6, the main protagonist in The Prisoner played by Patrick McGoohan, and portrayed by Jim Caviezelinthe remake.
• Six is a character in the television series Blossom played by Jenna von Oÿ.^[87]
• Six is the nickname of Kal Varrik, a central character in the television series Dark Matter, played by Roger Cross.^[88]
• Six is a History channel series that chronicles the operations and daily lives of SEAL Team Six.^[89]
• Six Feet Under, an HBO series that ran from 2005 to 2011.^[90]

• Number 6 (Teresa Palmer) is a character in the movie I Am Number Four (2011).^[91]
• The 6th Day (2000), starring Arnold Schwarzenegger.^[92]
• The Sixth Sense (1999), written and directed by M. Night Shyamalan and starring Haley Joel Osment and Bruce Willis.^[93]
• Girl 6 (1996), directed by Spike Lee.^[94]

• Six is a modern retelling of the lives of the six wives of Henry VIII presented as a pop concert.^[95]

• 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".
• Acoffin is traditionally buried six feet under the ground; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead^[96]
• There are said to be no more than six degrees of separation between any two people on Earth.^[97]
• In Western astrology, Virgo is the 6th astrological sign of the Zodiac^[98]
• The Six Dynasties form part of Chinese history^[99]
• Six is a lucky number in Chinese culture.^[100]
• The Birmingham Six were a British miscarriage of justice, held in prison for 16 years.^[101]
• "Six" is used as an informal slang term for the British Secret Intelligence Service, MI6.^[102]

• 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.^[103] Also, six is half a dozen.
• The maximum number of dots in a braille cell.^[104]
• Extrasensory perception is sometimes called the "sixth sense".^[105]
• Six Flags is an American company running amusement parks and theme parks in the U.S., Canada, and Mexico.^[106]
• 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".^[107] (For a similar example see also: Rainbow Six.)

• List of highways numbered 6
• Six degrees (disambiguation).

• 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

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