5 Explained

Number:	5
Numeral:	quinary
Prime:	3rd
Divisor:	1, 5
Roman:	V, v
Greek Prefix:	penta-/pent-
Latin Prefix:	quinque-/quinqu-/quint-
Lang1:	Greek
Lang1 Symbol:	ε (or Ε)
Lang2:	Arabic, Kurdish
Lang3:	Persian, Sindhi, Urdu
Lang4:	Ge'ez
Lang4 Symbol:	፭
Lang5:	Bengali
Lang6:	Kannada
Lang7:	Punjabi
Lang8:	Chinese numeral
Lang8 Symbol:	五
Lang9:	Armenian
Lang9 Symbol:	Ե
Lang10:	Devanāgarī
Lang11:	Hebrew
Lang12:	Khmer
Lang12 Symbol:	៥
Lang13:	Telugu
Lang14:	Malayalam
Lang15:	Tamil
Lang16:	Thai
Lang16 Symbol:	๕
Lang17:	Babylonian numeral
Lang18:	Egyptian hieroglyph, Chinese counting rod
Lang19:	Maya numerals
Lang20:	Morse code
Cardinal:	five
Ordinal:	5th(fifth)

5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number.

Humans, and many other animals, have 5 digits on their limbs.

Mathematics

Five is the third-smallest prime number,^[1] equal to the sum of the only consecutive positive integers to also be prime numbers (2 + 3). In integer sequences, five is also the second Fermat prime, and the third Mersenne prime exponent, as well as the fourth or fifth Fibonacci number;^[2] 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5).^[3]

In geometry, the regular five-sided pentagon is the first regular polygon that does not tile the plane with copies of itself, and it is the largest face that any of the five regular three-dimensional regular Platonic solid can have, as represented in the regular dodecahedron. For curves, a conic is determined using five points in the same way that two points are needed to determine a line.^[4]

In abstract algebra and the classification of finite simple groups, five is the count of exceptional Lie groups as well as the number of Mathieu groups that are sporadic groups. Five is also, more elementarily, the number of properties that are used to distinguish between the four fundamental number systems used in mathematics, which are rooted in the real numbers.

Aside from being the sum of the only consecutive positive integers to also be prime numbers, 2 + 3, it is also the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7);^{[5] [6]} this makes it the first balanced prime with equal-sized prime gaps above and below it (of 2).^[7]

5 is also the first safe prime^[8] where

(p - 1 )/ 2 for a prime p is also prime (2), and the first good prime, since it is the first prime number whose square (25) is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes (i.e., 3 × 7 = 21 and 11 × 2 = 22 are less than 25).^[9] 11, the fifth prime number, is the next good prime, that also forms the first pair of sexy primes with 5.^[10] 5 is the second Fermat prime of the form 2 ^{2 ^{n}} + 1 , of a total of five known Fermat primes.^[11]

Classes of integers

Wilson primes

5 is also the first of three known Wilson primes (5, 13, 563),^[12] where the square of a prime

p ^{2} divides (p -1 )! +1 . In the case of p = 5 ,( 5 - 1 )! + 1 = 4 ! + 1 = 24 + 1 = 25 = 5 ^{2} .

Perfect numbers

By the Euclid–Euler theorem, a

p of 5 in 2 ^{p-1} (2 ^{p} -1 ) yields the third perfect number, 496.^[13] Within the larger family of Ore numbers, 140 and 496, respectively the fourth and sixth indexed members, both contain a set of divisors that produce integer harmonic means equal to 5 (the only two such numbers).^{[14] [15]} Five is also the total number of known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors (the smallest such number is 6).^{[16] [17]}

In figurate numbers

5 is a pentagonal number in the sequence of figurate numbers, which starts: 1, 5, 12, 22, 35, ...^[18]

The fifth pentagonal and tetrahedral number is 35, which is equal to the sum of the first five triangular numbers: 1, 3, 6, 10, 15.^[19] In the sequence of pentatope numbers that start from the first (or fifth) cell of the fifth row of Pascal's triangle (left to right or from right to left), the first few terms are: 1, 5, 15, 35, 70, 126, 210, 330, 495, ...

As an odd number

Five is conjectured to be the only odd, untouchable number; if this is the case, then five will be the only odd prime number that is not the base of an aliquot tree.^[20]

Where five is the third prime number and odd number, every odd number greater than five is conjectured to be expressible as the sum of three prime numbers; Helfgott has provided a proof of this^[21] (also known as the odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes peer-review. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).^[22]

Other sequences

Powers

As a consequence of Fermat's little theorem and Euler's criterion, all squares are congruent to 0, 1, 4 (or −1) modulo 5.^[23] All integers

n \ge 34 can be expressed as the sum of five non-zero squares.^{[24] [25]}

Collatz conjecture

In the Collatz conjecture, 5 requires five steps to reach one by multiplying terms by three and adding one if the term is odd (starting with five itself),^[26] and dividing by two if they are even: ; the only other number to require five steps is 32 since 16 must be part of such path (see image to the right for a map of orbits for small odd numbers).^{[27] [28]}

Pisot–Vijayaraghavan numbers

\varphi (see for example, Binet's formula), 5 is strictly the fifth Fibonacci number (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...) — being the sum of 2 and 3 — as the only Fibonacci number greater than 1 that is equal to its position. In planar geometry, the ratio of a side and diagonal of a regular five-sided pentagon is also \varphi . Similarly, 5 is a member of the Perrin sequence, where 5 is both the fifth and sixth Perrin numbers, following (2, 3, 2) and preceding (7, 17);^[29] this sequence is instead associated with the plastic ratio, the least "small" Pisot–Vijayaraghavan number that does not supersede the golden ratio.^[30] This ratio is also associated with the Padovan sequence (1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, ...) where 5 is the twelfth member (and 12 the fifteenth), in-which the n −th Padovan number P(n) satisfies P( 0 ) = P( 1 ) = P( 2 ) = 1 , and P(n) = P(n -2 ) + P(n -3 ). ^[31] Manipulating Narayana's cows sequence N _{n} that has relations in proportion with the supergolden ratio as the fourth-smallest Pisot-Vijayaraghavan number whose value is less than the golden ratio, such that A _{n} = N _{n} + 2 N _{n-3} , five appears as the fourth member: (1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, ...).^{[32] [33]} On the other hand, 5 is part of the sequence of Pell numbers as the third indexed member, (0, 1, 2, 5, 12, 29, 70, 169, 408, ...).^[34] These numbers are approximately proportional to powers of the second-smallest Pisot Vijayaraghavan number following \varphi , the silver ratio \delta _{s} (and analogous to Fibonacci numbers, as powers of \varphi ), that appears in the regular octagon.

Permutation classes

There are five countably infinite Ramsey classes of permutations, where the age of each countable homogeneous permutation forms an individual Ramsey class

K of objects such that, for each natural number r and each choice of objects A,B \in K , there is no object C \in K where in any r -coloring of all subobjects of C isomorphic to A there exists a monochromatic subobject isomorphic to B .^[35] Aside from \{ 1 \} , the five classes of Ramsey permutations are the classes of:

Fraïssé limit

In general, the Fraïssé limit of a class

K of finite relational structure is the age of a countable homogeneous relational structure U if and only if five conditions hold for K : it is closed under isomorphism, it has only countably many isomorphism classes, it is hereditary, it is joint-embedded, and it holds the amalgamation property.

Magic figures

5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. Its

3 x 3 array has a magic constant M = 15 , where the sums of its rows, columns, and diagonals are all equal to fifteen.^[36] On the other hand, a normal 5 x 5 magic square has a magic constant M = 65 .^[37] 5 is also the value of the central cell the only non-trivial normal magic hexagon made of nineteen cells.^[38] The smallest magic star is a five-pointed magic pentagram, unique in that its smallest-possible magic constant M = 24 is only attainable using distinct integers.^[39]

Geometric properties

A pentagram, or five-pointed polygram, is the first proper star polygon constructed from the diagonals of a regular pentagon as self-intersecting edges that are proportioned in golden ratio,

\varphi . Where the equilateral triangle is the first proper regular polygon and only polygon without diagonals, the regular pentagon contains the same number of edges and diagonals.

The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol) appears prominently in Penrose tilings, and they are facets inside Kepler–Poinsot star polyhedra and Schläfli–Hess star polychora. A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges, often found inside Islamic Girih tiles (there are five different rudimentary types).^[40]

Graphs theory, and planar geometry

S _{5} of order 120 = 5!. For polynomial equations of degree and below can be solved with radicals, quintic equations of degree 5 and higher cannot generally be so solved (see, Abel–Ruffini theorem). This is related to the fact that the symmetric group S _{n} is a solvable group for n ⩽ 4 , and not for n ⩾ 5 .

The chromatic number of the plane is at least five, depending on the choice of set-theoretical axioms: the minimum number of colors required to color the plane such that no pair of points at a distance of 1 has the same color.^{[45] [46]} Whereas the hexagonal Golomb graph and the regular hexagonal tiling generate chromatic numbers of 4 and 7, respectively, a chromatic coloring of 5 can be attained under a more complicated graph where multiple four-coloring Moser spindles are linked so that no monochromatic triples exist in any coloring of the overall graph, as that would generate an equilateral arrangement that tends toward a purely hexagonal structure.

The plane also contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations: hexagonal, oblique, rectangular, centered rectangular, and square lattices. Uniform tilings of the plane, furthermore, are generated from combinations of only five regular polygons: the triangle, square, hexagon, octagon, and the dodecagon.^[47] The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.^[48]

Polyhedral geometry

There are five Platonic solids in three-dimensional space that are regular: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.^[49] The dodecahedron in particular contains pentagonal faces, while the icosahedron, its dual polyhedron, has a vertex figure that is a regular pentagon. These five regular solids are responsible for generating thirteen figures that classify as semi-regular, which are called the Archimedean solids. There are also five:

Moreover, there are also precisely five uniform prisms and antiprisms that contain pentagons or pentagrams as faces — the pentagonal prism and antiprism, and the pentagrammic prism, antiprism, and crossed-antiprism.^[50]

Four-dimensional space

The pentatope, or 5-cell, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry

A _{4} of order 120 = 5! and S _{5} group structure. Made of five tetrahedra, its Petrie polygon is a regular pentagon and its orthographic projection is equivalent to the complete graph K₅. It is one of six regular 4-polytopes, made of thirty-one elements: five vertices, ten edges, ten faces, five tetrahedral cells and one 4-face.^[51]

Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora based on only twenty-five uniform polyhedra:

A _{4} , B _{4} , D _{4} , F _{4} , and H _{4} , accompanied by a fifth or sixth general group of unique 4-prisms of Platonic and Archimedean solids. There are also a total of five Coxeter groups that generate non-prismatic Euclidean honeycombs in 4-space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite facets, as with the order-5 5-cell honeycomb and the order-5 120-cell honeycomb, both of which have five cells around each face. Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10. Likewise, analogues of four-dimensional

H _{4} hexadecachoric or F _{4} icositetrachoric symmetry do not exist in dimensions n ⩾ 5 ; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have H _{4} and F _{4} symmetry. There are also five regular projective 4-polytopes in the fourth dimension, all of which are hemi-polytopes of the regular 4-polytopes, with the exception of the 5-cell.^[52] Only two regular projective polytopes exist in each higher dimensional space.

Generally, star polytopes that are regular only exist in dimensions

2 ⩽ n < 5 , and can be constructed using five Miller rules for stellating polyhedra or higher-dimensional polytopes.^[53]

Five-dimensional space

The 5-simplex or hexateron is the five-dimensional analogue of the 5-cell, or 4-simplex. It has Coxeter group

A _{5} as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group S _{6} , the only finite symmetric group which has an outer automorphism. The 5-cube, made of ten tesseracts and the 5-cell as its vertex figure, is also regular and one of thirty-one uniform 5-polytopes under the Coxeter B _{5} hypercubic group. The demipenteract, with one hundred and twenty cells, is the only fifth-dimensional semi-regular polytope, and has the rectified 5-cell as its vertex figure, which is one of only three semi-regular 4-polytopes alongside the rectified 600-cell and the snub 24-cell. In the fifth dimension, there are five regular paracompact honeycombs, all with infinite facets and vertex figures; no other regular paracompact honeycombs exist in higher dimensions.^[54] There are also exclusively twelve complex aperiotopes in C ^{n} complex spaces of dimensions n  ⩾ 5 ; alongside complex polytopes in C ^{5} and higher under simplex, hypercubic and orthoplex groups (with van Oss polytopes).^[55]

Veronese surface

P ^{5} generalizes a linear condition \nu: P ^{2\to} P ^{5} for a point to be contained inside a conic, where five points determine a conic.

In finite simple groups

Lie groups

There are five complex exceptional Lie algebras:

ak{g} _{2} , ak{f} _{4} , ak{e} _{6} , ak{e} _{7} , and ak{e} _{8} . The smallest of these,

ak{g} _{2} of real dimension 28, can be represented in five-dimensional complex space and projected as a ball rolling on top of another ball, whose motion is described in two-dimensional space.^[56] ak{e} _{8} is the largest, and holds the other four Lie algebras as subgroups, with a representation over R in dimension 496. It contains an associated lattice that is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the 600-cell, whose Euclidean norms define a quadratic form on a lattice structure isomorphic to the optimal configuration of spheres in eight dimensions.^[57] This sphere packing E _{8} lattice structure in 8-space is held by the vertex arrangement of the 5₂₁ honeycomb, one of five Euclidean honeycombs that admit Gosset's original definition of a semi-regular honeycomb, which includes the three-dimensional alternated cubic honeycomb.^{[58] [59]} The smallest simple isomorphism found inside finite simple Lie groups is A _{5} \cong A _{1} ( 4 ) \cong A _{1} ( 5 ) ,^[60] where here A _{n} represents alternating groups and A _{n} (q) classical Chevalley groups. In particular, the smallest non-solvable group is the alternating group on five letters, which is also the smallest simple non-abelian group.

Sporadic groups

Mathieu groups

The five Mathieu groups constitute the first generation in the happy family of sporadic groups. These are also the first five sporadic groups to have been described, defined as

M _{n} multiply transitive permutation groups on n objects, with n ∈ .^[61] In particular, M _{11} , the smallest of all sporadic groups, has a rank 3 action on fifty-five points from an induced action on unordered pairs, as well as two five-dimensional faithful complex irreducible representations over the field with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with n elements.^[62] Of precisely five different conjugacy classes of maximal subgroups of M _{11} , one is the almost simple symmetric group S _{5} (of order 5!), and another is M _{10} , also almost simple, that functions as a point stabilizer which contains five as its largest prime factor in its group order: . On the other hand, whereas M _{11} is sharply 4-transitive, M _{12} is sharply 5-transitive and M _{24} is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups.^[63] M _{22} has the first five prime numbers as its distinct prime factors in its order of ; all Mathieu groups are subgroups of M _{24} , which under the Witt design W _{24} of Steiner system \operatorname{S( 5 , 8 , 24 )} emerges a construction of the extended binary Golay code B _{24} that has M _{24} as its automorphism group. W _{24} generates octads from code words of Hamming weight 8 from the extended binary Golay code, one of five different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24. The Witt design and the extended binary Golay code in turn can be used to generate a faithful construction of the 24-dimensional Leech lattice Λ₂₄, which is primarily constructed using the Weyl vector ( 0 , 1 , 2 , 3 , ... , 24 ; 70 ) that admits the only non-unitary solution to the cannonball problem, where the sum of the squares of the first twenty-four integers is equivalent to the square of another integer, the fifth pentatope number (70). The subquotients of the automorphism of the Leech lattice, Conway group Co _{0} , is in turn the subject of the second generation of seven sporadic groups.

Harada-Norton group

F _{1} arises from the product between Harada–Norton sporadic group HN and a group of order 5.^{[64] [65]} On its own, HN can be represented using standard generators (a,b,ab) that further dictate a condition where o([a, b]) = 5 .^{[66] [67]} This condition is also held by other generators that belong to the Tits group T ,^[68] the only finite simple group that is a non-strict group of Lie type that can also classify as sporadic. Furthermore, over the field with five elements, HN holds a 133-dimensional representation where 5 acts on a commutative yet non-associative product as a 5-modular analogue of the Griess algebra V _{2} ,^[69] which holds F _{1} as its automorphism group.

List of basic calculations

Division	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
5 ÷ x	5	2.5	1.	1.25	1	0.8	0.	0.625	0.	0.5		0.	0.41	0.	0.3	0.
x ÷ 5	0.2	0.4	0.6	0.8		1.2	1.4	1.6	1.8	2		2.2	2.4	2.6	2.8	3

Exponentiation	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
5	5	25	125	625	3125	15625	78125	390625	1953125	9765625		48828125	244140625	1220703125	6103515625	30517578125
x	1	32	243	1024		7776	16807	32768	59049	100000		161051	248832	371293	537824	759375

Decimal properties

All multiples of 5 will end in either 5 or, and vulgar fractions with 5 or in the denominator do not yield infinite decimal expansions because they are prime factors of 10, the base.

In the powers of 5, every power ends with the number five, and from 5³ onward, if the exponent is odd, then the hundreds digit is 1, and if it is even, the hundreds digit is 6.

A number

n raised to the fifth power always ends in the same digit as n .

Evolution of the Arabic digit

The evolution of the modern Western digit for the numeral for five is traced back to the Indian system of numerals, where on some earlier versions, the numeral bore resemblance to variations of the number four, rather than "5" (as it is represented today). The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. Later on, Arabic traditions transformed the digit in several ways, producing forms that were still similar to the numeral for four, with similarities to the numeral for three; yet, still unlike the modern five.^[70] It was from those digits that Europeans finally came up with the modern 5 (represented in writings by Dürer, for example).

While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in .

On the seven-segment display of a calculator and digital clock, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa. It is one of three numbers, along with 4 and 6, where the number of segments matches the number.

Other fields

Astronomy

There are five Lagrangian points in a two-body system.

Biology

There are usually considered to be five senses (in general terms); the five basic tastes are sweet, salty, sour, bitter, and umami.^[71] Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity. Five is the number of appendages on most starfish, which exhibit pentamerism.^[72]

Computing

5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.^[73]

Literature

Poetry

A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.^[74]

Music

Modern musical notation uses a musical staff made of five horizontal lines.^[75] A scale with five notes per octave is called a pentatonic scale.^[76] A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.^[77] In harmonics, the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A major triad chord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third.

Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.

Religion

Judaism

The Book of Numbers is one of five books in the Torah; the others being the books of Genesis, Exodus, Leviticus, and Deuteronomy. They are collectively called the Five Books of Moses, the Pentateuch (Greek for "five containers", referring to the scroll cases in which the books were kept), or Humash (Hebrew: חומש, Hebrew for "fifth").^[78] The Khamsa, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by Jews; that same symbol is also very popular in Arabic culture, known to protect from envy and the evil eye.^[79]

Christianity

There are traditionally five wounds of Jesus Christ in Christianity: the nail wounds in Christ's two hands, the nail wounds in Christ's two feet, and the Spear Wound of Christ (respectively at the four extremities of the body, and the head).^[80]

Islam

The Five Pillars of Islam.^[81]

Mysticism

Gnosticism

The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.

Alchemy

According to ancient Greek philosophers such as Aristotle, the universe is made up of five classical elements: water, earth, air, fire, and ether. This concept was later adopted by medieval alchemists and more recently by practitioners of Neo-Pagan religions such as Wicca. There are five elements in the universe according to Hindu cosmology: (earth, fire, water, air and space, respectively). In East Asian tradition, there are five elements: water, fire, earth, wood, and metal.^[82] The Japanese names for the days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the five planets visible with the naked eye.^[83] Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday. There are also five elements in the traditional Chinese Wuxing.^[84]

Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth), as a union of these.^[85] The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.

Miscellaneous fields

• "Give me five" is a common phrase used preceding a high five.
• The Olympic Games have five interlocked rings as their symbol, representing the number of inhabited continents represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).^[86]
• The number of dots in a quincunx.^[87]

See also

5 (disambiguation)

References

Further reading

• Book: Wells, D. . The Penguin Dictionary of Curious and Interesting Numbers . The Penguin Dictionary of Curious and Interesting Numbers . London, UK . . 1987 . 58–67.

External links

• Prime curiosities: 5

Notes and References

1. Web site: Weisstein. Eric W. . 5 . 2020-07-30 . mathworld.wolfram.com . en.
2. Web site: Weisstein. Eric W. . Twin Primes . 2020-07-30 . mathworld.wolfram.com . en.
3. A003273 . Congruent numbers. 2016-06-01.
4. A. C. . Dixon . Alfred Cardew Dixon . The Conic through Five Given Points . The Mathematical Gazette . 4 . 70 . March 1908 . 228–230 . The Mathematical Association . 3605147 . 10.2307/3605147 . 125356690 .
5. 2023-02-14.
6. 2023-02-14.
7. 2023-02-14.
8. 2023-02-14.
9. 2016-06-01.
10. 2023-01-14.
11. 2022-07-21.
12. 2023-09-06 .
13. 2022-10-13 .
14. 2022-12-26 .
15. 2022-12-26 .
16. Book: Richard K. Guy. Richard K. Guy. Unsolved Problems in Number Theory. Springer-Verlag. 2004. 0-387-20860-7 . 84–86.
17. 2023-01-10 .
18. 2022-11-08 .
19. 2022-11-08 . In general, the sum of n consecutive triangular numbers is the nth tetrahedral number.
20. Pomerance. Carl. Yang. Hee-Sung . 14 June 2012. On Untouchable Numbers and Related Problems. Dartmouth College. 1. 30344483 . math.dartmouth.edu. 2010 Mathematics Subject Classification. 11A25, 11Y70, 11Y16.
21. Book: Helfgott, Harald Andres . Jang . Sun Young . 2014 . The ternary Goldbach problem . https://www.imj-prg.fr/wp-content/uploads/2020/prix/helfgott2014.pdf . Seoul International Congress of Mathematicians Proceedings . 2 . Kyung Moon SA . Seoul, KOR . 391–418 . 978-89-6105-805-6 . 913564239 .
22. Tao . Terence . March 2014 . Every odd number greater than 1 has a representation is the sum of at most five primes . Mathematics of Computation . 83 . 286 . 997–1038 . 10.1090/S0025-5718-2013-02733-0 . 3143702 . 2618958 .
23. James A. . Sellers . An unexpected congruence modulo 5 for 4-colored generalized Frobenius partitions . J. Indian Math. Soc. . New Series . 2013 . Special Issue . 99 . . Pune, IMD . 2013arXiv1302.5708S . 1302.5708 . 157339 . 1290.05015 . 116931082 .
24. Book: Niven . Ivan . Ivan M. Niven . Zuckerman . Herbert S. . Montgomery . Hugh L. . Hugh Lowell Montgomery . An Introduction to the Theory of Numbers . . New York, NY . 5th . 1980 . 144, 145 . 978-0-19-853171-5 .
25. 2023-09-20 .

    Only twelve integers up to 33 cannot be expressed as the sum of five non-zero squares: where 2, 3 and 7 are the only such primes without an expression.

26. Web site: Sloane . N. J. A. . Neil Sloane . 3x+1 problem . . The OEIS Foundation . 2023-01-24 .
27. 2023-01-24 .
28. Lagarias . Jeffrey C. . Jeffrey Lagarias . The 3x + 1 Problem and Its Generalizations . . 92 . 1 . . January 1985 . 4–6. 10.1080/00029890.1985.11971528 . 2322189 .
29. Web site: Weisstein. Eric W.. Perrin Sequence. 2020-07-30. mathworld.wolfram.com. en.
30. Book: M.J. Bertin . A. Decomps-Guilloux . M. Grandet-Hugot . M. Pathiaux-Delefosse . J.P. Schreiber . Pisot and Salem Numbers . Birkhäuser . 1992 . 3-7643-2648-4 .
31. 2024-05-09 .
32. 2024-05-26 .
33. 2024-05-09 .
34. 2024-05-09 .
35. Böttcher . Julia. Julia Böttcher . Foniok . Jan . Ramsey Properties of Permutations . The Electronic Journal of Combinatorics . 20 . 1 . 2013 . P2 . 10.37236/2978 . 1103.5686v2 . 17184541 . 1267.05284 .
36. Web site: William H. Richardson . Magic Squares of Order 3 . Wichita State University Dept. of Mathematics . 2022-07-14 .
37. 2024-08-14 .
38. A Unique Magic Hexagon . Trigg, C. W. . Recreational Mathematics Magazine . February 1964 . 2022-07-14 .
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41. Burnstein . Michael. Kuratowski-Pontrjagin theorem on planar graphs . . Series B . 24 . 2 . 1978 . 228–232 . 10.1016/0095-8956(78)90024-2 . free .
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43. Noga Alon . Alon . Noga . Grytczuk . Jaroslaw . Hałuszczak . Mariusz . Riordan . Oliver . Nonrepetitive colorings of graphs . Random Structures & Algorithms . 2 . 3–4 . 2002 . 337 . 10.1002/rsa.10057 . 5724512 . 1945373 . A coloring of the set of edges of a graph G is called non-repetitive if the sequence of colors on any path in G is non-repetitive...In Fig. 1 we show a non-repetitive 5-coloring of the edges of P... Since, as can easily be checked, 4 colors do not suffice for this task, we have π(P) = 5. .
44. Web site: Royle . G. . etal . February 2001 . Cubic Symmetric Graphs (The Foster Census) . https://web.archive.org/web/20080720005020/http://www.cs.uwa.edu.au/~gordon/remote/foster/ . 2008-07-20.
45. de Grey . Aubrey D.N.J. . Aubrey de Grey . The Chromatic Number of the Plane is At Least 5 . . 28 . 5–18 . 2018 . 1804.02385 . 3820926 . 119273214 .
46. Exoo . Geoffrey . Ismailescu . Dan . The Chromatic Number of the Plane is At Least 5: A New Proof . . 64 . 216–226 . . New York, NY . 2020 . 10.1007/s00454-019-00058-1 . 1805.00157 . 4110534 . 119266055 . 1445.05040 .
47. Branko . Grünbaum . Branko Grünbaum . Geoffrey . Shepard . G.C. Shephard . Tilings by Regular Polygons . November 1977 . . 50 . 5 . Taylor & Francis, Ltd.. 227–236 . 10.2307/2689529 . 2689529 . 123776612 . 0385.51006 .
48. Book: Grünbaum . Branko . Branko Grünbaum . Shephard . Geoffrey C. . Tilings and Patterns . New York . W. H. Freeman and Company . 1987 . 978-0-7167-1193-3 . Tilings by polygons . 0857454 . registration . Section 9.3: "Other Monohedral tilings by convex polygons".
49. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61
50. Har'El . Zvi . Uniform Solution for Uniform Polyhedra . . 47 . 57–110 . . Netherlands . 1993 . 10.1007/BF01263494 . 1230107 . 0784.51020 . 120995279 .

    "In tables 4 to 8, we list the seventy-five nondihedral uniform polyhedra, as well as the five pentagonal prisms and antiprisms, grouped by generating Schwarz triangles."
    Appendix II: Uniform Polyhedra.

51. Book: H. S. M. Coxeter . Harold Scott MacDonald Coxeter . . . 3rd . 1973 . New York . 1–368 . 978-0-486-61480-9 .
52. Book: McMullen . Peter . Peter McMullen . Schulte . Egon . Egon Schulte . Abstract Regular Polytopes . registration . Cambridge University Press . Cambridge . Encyclopedia of Mathematics and its Applications . 92 . 2002 . 162–164 . 10.1017/CBO9780511546686 . 0-521-81496-0 . 1965665 . 115688843 .
53. Book: Coxeter . H. S. M. . du Val . P. . Flather . H. T. . Petrie . J.F. . H. S. M. Coxeter . Patrick du Val . 2 . The Fifty-Nine Icosahedra . . 1st . New York . 1982 . 7, 8 . 10.1007/978-1-4613-8216-4 . 978-0-387-90770-3 . 8667571 . 118322641.
54. Web site: H.S.M. Coxeter. Regular Honeycombs in Hyperbolic Space . 1956. 168. 10.1.1.361.251 .
55. Book: H. S. M. Coxeter . H. S. M. Coxeter . Regular Complex Polytopes . Cambridge University Press . 2nd . 1991 . 144–146 . 10.2307/3617711 . 3617711 . 978-0-521-39490-1 . 116900933 . 0732.51002 .
56. G₂ and the rolling ball . Baez . John C. . John C. Baez. Huerta . John . Trans. Amer. Math. Soc. . 366 . 10 . 2014 . 5257–5293 . 10.1090/s0002-9947-2014-05977-1 . 3240924 . 50818244 . free .
57. Baez . John C. . John C. Baez . From the Icosahedron to E₈ . London Math. Soc. Newsletter . 476 . 18–23 . 2018 . 1712.06436 . 3792329 . 119151549 . 1476.51020 .
58. H. S. M. Coxeter . H. S. M. Coxeter . Seven Cubes and Ten 24-Cells . . 19 . 1998 . 2 . 156–157 . 10.1007/PL00009338 . free . 0898.52004 . 206861928 .
59. Thorold Gosset . Thorold Gosset . On the regular and semi-regular figures in space of n dimensions . Messenger of Mathematics . 29 . 1900 . 43–48 . 30.0494.02 .
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61. Book: Robert L. Griess, Jr. . Robert Griess . Twelve Sporadic Groups . Springer Monographs in Mathematics . Springer-Verlag . Berlin . 1998 . 1−169 . 978-3-540-62778-4 . 10.1007/978-3-662-03516-0 . 1707296 . 116914446 . 0908.20007 .
62. Jansen . Christoph . 2005 . The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups . . 8 . 123–124 . . 10.1112/S1461157000000930 . free . 2153793 . 121362819 . 1089.20006 .
63. Book: Cameron, Peter J. . Projective and Polar Spaces . Chapter 9: The geometry of the Mathieu groups . https://webspace.maths.qmul.ac.uk/p.j.cameron/pps/pps9.pdf . University of London, Queen Mary and Westfield College . 1992 . 139. 978-0-902-48012-4 . 115302359 .
64. Lux . Klaus . Noeske . Felix . Ryba . Alexander J. E. . The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2 . . . 319 . 1 . 2008 . Amsterdam . 320–335 . 10.1016/j.jalgebra.2007.03.046 . free . 2378074 . 120706746 . 1135.20007 .
65. Wilson . Robert A. . Robert Arnott Wilson . The odd local subgroups of the Monster . Journal of Australian Mathematical Society (Series A) . . 44 . 1 . 12–13 . 2009 . Cambridge . 10.1017/S1446788700031323 . free . 914399 . 123184319 . 0636.20014 .
66. Book: Wilson, R.A . Robert Arnott Wilson . The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249) . An Atlas of Sporadic Group Representations . https://webspace.maths.qmul.ac.uk/r.a.wilson/pubs_files/ASGRweb.pdf . Cambridge University Press . Cambridge . 1998 . 267 . 10.1017/CBO9780511565830.024 . 978-0-511-56583-0 . 726827806 . 0914.20016 . 59394831 .
67. Nickerson . S.J. . Wilson . R.A. . Robert Anton Wilson . Semi-Presentations for the Sporadic Simple Groups . Experimental Mathematics . 14 . 3 . 367 . . 2011 . Oxfordshire . 10.1080/10586458.2005.10128927 . 2172713 . 1087.20025 . 13100616 .
68. Web site: Wilson . R.A. . Robert Arnott Wilson . Parker . R.A. . Richard A. Parker . Nickerson . S.J. . Bray . J.N. . Exceptional group ²F₄(2)', Tits group T . ATLAS of Finite Group Representations . 1999.
69. Ryba . A. J. E. . A natural invariant algebra for the Harada-Norton group . . . 119 . 4 . Cambridge . 1996 . 597–614 . 10.1017/S0305004100074454 . 1996MPCPS.119..597R . 1362942 . 119931824 . 0851.20034 .
70. 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): 394, Fig. 24.65
71. Book: Marcus, Jacqueline B.. Culinary Nutrition: The Science and Practice of Healthy Cooking. 2013-04-15. Academic Press. 978-0-12-391883-3. 55. en. There are five basic tastes: sweet, salty, sour, bitter and umami....
72. Book: Cinalli. G.. Pediatric Hydrocephalus. Maixner. W. J.. Sainte-Rose. C.. 2012-12-06. Springer Science & Business Media. 978-88-470-2121-1. 19. en. The five appendages of the starfish are thought to be homologous to five human buds.
73. Book: Pozrikidis, Constantine. XML in Scientific Computing. 2012-09-17. CRC Press. 978-1-4665-1228-3. 209. en. 5 5 005 ENQ (enquiry).
74. Book: Veith (Jr.). Gene Edward. Omnibus IV: The Ancient World. Wilson. Douglas. 2009. Veritas Press. 978-1-932168-86-0. 52. en. The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter).
75. Web site: STAVE meaning in the Cambridge English Dictionary. 2020-08-02. dictionary.cambridge.org. en. the five lines and four spaces between them on which musical notes are written.
76. Book: Ricker, Ramon. Pentatonic Scales for Jazz Improvisation. 1999-11-27. Alfred Music. 978-1-4574-9410-9. 2. en. Pentatonic scales, as used in jazz, are five note scales.
77. Book: Danneley, John Feltham. An Encyclopaedia, Or Dictionary of Music ...: With Upwards of Two Hundred Engraved Examples, the Whole Compiled from the Most Celebrated Foreign and English Authorities, Interspersed with Observations Critical and Explanatory. 1825. editor, and pub.. en. are the perfect fourth, perfect fifth, and the octave.
78. Web site: Pelaia. Ariela. Judaism 101: What Are the Five Books of Moses?. 2020-08-03. Learn Religions. en.
79. Book: Zenner, Walter P.. Persistence and Flexibility: Anthropological Perspectives on the American Jewish Experience. 1988-01-01. SUNY Press. 978-0-88706-748-8. 284. en.
80. Web site: CATHOLIC ENCYCLOPEDIA: The Five Sacred Wounds. 2020-08-02. www.newadvent.org.
81. Web site: PBS – Islam: Empire of Faith – Faith – Five Pillars. 2020-08-03. www.pbs.org.
82. Book: Yoon, Hong-key. The Culture of Fengshui in Korea: An Exploration of East Asian Geomancy. 2006. Lexington Books. 978-0-7391-1348-6. 59. en. The first category is the Five Agents [Elements] namely, Water, Fire, Wood, Metal, and Earth..
83. Book: Walsh, Len. 2008-11-15 . Read Japanese Today: The Easy Way to Learn 400 Practical Kanji . Tuttle Publishing. 978-1-4629-1592-7. en . The Japanese names of the days of the week are taken from the names of the seven basic nature symbols.
84. Chen . Yuan . 2014 . Legitimation Discourse and the Theory of the Five Elements in Imperial China . Journal of Song-Yuan Studies . en . 44 . 1 . 325–364 . 10.1353/sys.2014.0000 . 147099574 . 2154-6665.
85. Book: Kronland-Martinet . Richard . Computer Music Modeling and Retrieval. Sense of Sounds: 4th International Symposium, CMMR 2007, Copenhagen, Denmark, August 2007, Revised Papers . Ystad . Sølvi . Jensen . Kristoffer . 2008-07-19 . Springer . 978-3-540-85035-9 . 502 . en . Plato and Aristotle postulated a fifth state of matter, which they called "idea" or quintessence" (from "quint" which means "fifth").
86. Web site: 2020-06-23. Olympic Rings – Symbol of the Olympic Movement. 2020-08-02. International Olympic Committee. en.
87. Book: Laplante, Philip A.. Comprehensive Dictionary of Electrical Engineering. 2018-10-03. CRC Press. 978-1-4200-3780-7. 562. en. quincunx five points.