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 number25.
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]
It is the first primitivepseudoperfect number,[8] and all integers that are multiples of 6 are pseudoperfect (all multiples of a perfect number are pseudoperfect); six is also the smallest Granville number, or -perfect number.[9]
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).[15]
There is not a prime such that the multiplicative order of 2 modulo is 6, that is,
By Zsigmondy's theorem, if is a natural number that is not 1 or 6, then there is a prime such that . See A112927 for such .
The ring of integer of the sixth cyclotomic field Q(ζ6), which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where .
The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.[16]
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.[20] The densest sphere packing of the plane is obtained by extending this pattern to the hexagonallattice in which each circle touches just six others.
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:[22]
where is the surface area of the sphere, the volume of the object, and the surface area of the object.
, 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 for which there is a construction of isomorphic objects on an -set, invariant under all permutations of , but not naturally in one-to-one correspondence with the elements of . This can also be expressed category theoretically: consider the category whose objects are the element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for .
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.[30][31][32][33] 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.[34]
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 .[35]
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.
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
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.[47]
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."
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.[68]
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.[74]
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.)[75]
Six players from each team on each side play against each other.[76]
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.[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".[79]
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.)[80]
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
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.
The maximum number of dots in a braille cell.[103]
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.)
^Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.66
^"CHURCH FATHERS: City of God, Book XI (St. Augustine)". www.newadvent.org. Retrieved 2020-08-04. These works are recorded to have been completed in six days (the same day being six times repeated), because six is a perfect number
^Bary, William Theodore De; DeBary, William T.; Chan, Wing-tsit; Lufrano, Richard; Ching, Julia; Johnson, David; Liu, Kwang-Ching; Mungello, David (1999). Sources of Chinese Tradition. Columbia University Press. ISBN978-0-231-11270-3. ...and the Six Ministries were made...
^Sedgwick, Marcus (2011-07-05). White Crow. Roaring Brook Press. p. 145. ISBN978-1-4299-7634-3. The cells of honeycombs are six-sided because a hexagon is the most material-efficient tessellation
^Sports, The National Alliance For Youth (2009-05-11). Coaching Volleyball For Dummies. John Wiley & Sons. p. 48. ISBN978-0-470-53398-7. In a regulation volleyball match with six players on each side of the court,
^Wilkinson, Endymion Porter; Wilkinson, Scholar and Diplomat (Eu Ambassador to China 1994-2001) Endymion (2000). Chinese History: A Manual. Harvard Univ Asia Center. p. 11. ISBN978-0-674-00249-4.((cite book)): CS1 maint: numeric names: authors list (link)