5
| ||||
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−1 0 1 2 3 4 5 6 7 8 9 →
← 0 10 20 30 40 50 60 70 80 90 → | ||||
Cardinal | five | |||
Ordinal | 5th (fifth) | |||
Numeral system | quinary | |||
Factorization | prime | |||
Prime | 3rd | |||
Divisors | 1, 5 | |||
Greek numeral | Ε´ | |||
Roman numeral | V, v | |||
Greek prefix | penta-/pent- | |||
Latin prefix | quinque-/quinqu-/quint- | |||
Binary | 1012 | |||
Ternary | 123 | |||
Senary | 56 | |||
Octal | 58 | |||
Duodecimal | 512 | |||
Hexadecimal | 516 | |||
Greek | ε (or Ε) | |||
Arabic, Kurdish | ٥ | |||
Persian, Sindhi, Urdu | ۵ | |||
Ge'ez | ፭ | |||
Bengali | ৫ | |||
Kannada | ೫ | |||
Punjabi | ੫ | |||
Chinese numeral | 五 | |||
Armenian | Ե | |||
Devanāgarī | ५ | |||
Hebrew | ה | |||
Khmer | ៥ | |||
Telugu | ౫ | |||
Malayalam | ൫ | |||
Tamil | ௫ | |||
Thai | ๕ | |||
Babylonian numeral | 𒐙 | |||
Egyptian hieroglyph, Chinese counting rod | ||||| | |||
Maya numerals | 𝋥 | |||
Morse code | ..... |
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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Pythagoras%27_Special_Triples.svg/120px-Pythagoras%27_Special_Triples.svg.png)
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.
Historically, 5 has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
Number systems
In the classification of number systems, the real numbers and its three subsequent Cayley–Dickson constructions of algebras over the field of the real numbers (i.e. the complex numbers , the quaternions , and the octonions ) are normed division algebras that hold up to five different principal algebraic properties of interest: whether the algebras are ordered, and whether they hold commutative, associative, alternative, and power-associative multiplicative properties.[5] Whereas the real numbers contain all five properties, the octonions are only alternative and power-associative. In comparison, the sedenions , which represent a fifth algebra in this series, is not a composition algebra unlike and , is only power-associative, and is the first algebra to contain non-trivial zero divisors as with all further algebras over larger fields.[6] Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16.
Classes of integers
Five is the third prime number, and more specifically, the second super-prime since its prime index is prime.[1] 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);[7][8] this makes it the first balanced prime with equal-sized prime gaps above and below it (of 2).[9]
5 is also the first safe prime[10] where for a prime 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).[11] 11, the fifth prime number, is the next good prime, that also forms the first pair of sexy primes with 5.[12]
Mersenne primes
5 is the third Mersenne prime exponent for , which yields the eleventh prime number and fifth super-prime 31.[13][14][1] This is the prime index of the third Mersenne prime and second double Mersenne prime 127,[15] as well as the third double Mersenne prime exponent for the number 2,147,483,647,[15] which is the largest value that a signed 32-bit integer field can hold. Collectively, 5 and 31 generate a sum of 36 (the square of 6) and a difference of 26, which is the only number to lie between a square and a cube (respectively, 25 and 27).[16]
Fermat primes
5 is the second Fermat prime of the form , and more generally the second Sierpiński number of the first kind, .[17] There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.[18] Combinations from these five Fermat primes generate thirty-one polygons with an odd number of sides that can be constructed purely with a compass and straight-edge, which includes the five-sided regular pentagon.[19][20]: pp.137–142 Apropos, thirty-one is also equal to the sum of the maximum number of areas inside a circle that are formed from the sides and diagonals of the first five -sided polygons, which is equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem.[21][20]: pp.76–78
Wilson primes
5 is also the first of three known Wilson primes (5, 13, 563),[22] where the square of a prime divides In the case of ,
The first two Wilson primes are also consecutive Proth primes[23] and Markov numbers, where 5 appears in solutions to the Markov Diophantine equations: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... (OEIS: A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5). 5 is also the third factorial prime,[24] since , and the first non-trivial alternating factorial equal to the absolute value of the alternating sum of the first three factorials, [25]
Perfect numbers
The sums of the first five non-primes greater than zero 1 + 4 + 6 + 8 + 9 and the first five prime numbers 2 + 3 + 5 + 7 + 11 both equal 28; the seventh triangular number and like 6 a perfect number, which also includes 496, the thirty-first triangular number and perfect number of the form () with a of , by the Euclid–Euler theorem.[26][27][28] 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.[29][30] The fifth Mersenne prime, 8191,[14] splits into 4095 and 4096, with the latter being the fifth superperfect number[31] and the sixth power of four, 46. Five is also the total number of known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors.[32][33] The smallest such number is 6.
The factorial of five is multiply perfect like 28 and 496.[34] It is the sum of the first fifteen non-zero positive integers and 15th triangular number, which in-turn is the sum of the first five non-zero positive integers and 5th triangular number. Furthermore, , where 125 is the second number to have an aliquot sum of 31 (after the fifth power of two, 32).[35]
In figurate numbers
5 is a pentagonal number in the sequence of figurate numbers, which starts: 1, 5, 12, 22, 35, ...[36]
- 5 is a centered tetrahedral number: 1, 5, 15, 35, 69, ...[37] Every centered tetrahedral number with an index of 2, 3 or 4 modulo 5 is divisible by 5.
- 5 is a square pyramidal number: 1, 5, 14, 30, 55, ...[38] The first four members add to 50 while the fifth indexed member in the sequence is 55.
- 5 is a centered square number: 1, 5, 13, 25, 41, ...[39] The fifth square number or 52 is 25, which features in the proportions of the two smallest (3, 4, 5) and (5, 12, 13) primitive Pythagorean triples.[40]
31 is the first prime centered pentagonal number,[41] and the fifth centered triangular number.[42] 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.[43] 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, ...[44]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Pentagon_subdivisions.svg/220px-Pentagon_subdivisions.svg.png)
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.[45]
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[46] (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).[47]
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.[48] All integers can be expressed as the sum of five non-zero squares.[49][50]
Regarding Waring's problem, , where every natural number is the sum of at most thirty-seven fifth powers.[51][52]
Pisot–Vijayaraghavan numbers
In the Fibonacci sequence, which can be defined in terms of the golden ratio (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 — [1] 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 . 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);[53] this sequence is instead associated with the plastic ratio, the least "small" Pisot–Vijayaraghavan number that does not supersede the golden ratio.[54] 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 −th Padovan number satisfies and [55] Manipulating Narayana's cows sequence 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 , five appears as the fourth member: (1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, ...).[56][57] 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, ...).[58] These numbers are approximately proportional to powers of the second-smallest Pisot Vijayaraghavan number following , the silver ratio (and analogous to Fibonacci numbers, as powers of ), 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 of objects such that, for each natural number and each choice of objects , there is no object where in any -coloring of all subobjects of isomorphic to there exists a monochromatic subobject isomorphic to .[59]: pp.1, 2 Aside from , the five classes of Ramsey permutations are the classes of:[59]: p.4
- Identity permutations, and reversals
- Increasing sequences of decreasing sequences, and decreasing sequences of increasing sequences
- All permutations
Fraïssé limit
In general, the Fraïssé limit of a class of finite relational structure is the age of a countable homogeneous relational structure if and only if five conditions hold for : 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.[59]: p.3
Magic figures
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/af/Magic_Square_Lo_Shu.svg/120px-Magic_Square_Lo_Shu.svg.png)
5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. Its array has a magic constant of , where the sums of its rows, columns, and diagonals are all equal to fifteen.[60] On the other hand, a normal magic square[a] has a magic constant of , where 5 and 13 are the first two Wilson primes.[22] The fifth number to return for the Mertens function is 65,[61] with counting the number of square-free integers up to with an even number of prime factors, minus the count of numbers with an odd number of prime factors. 65 is the nineteenth biprime with distinct prime factors,[62] with an aliquot sum of 19 as well[35] and equivalent to 15 + 24 + 33 + 42 + 51.[63] It is also the magic constant of the Queens Problem for ,[64] the fifth octagonal number,[65] and the Stirling number of the second kind that represents sixty-five ways of dividing a set of six objects into four non-empty subsets.[66] 13 and 5 are also the fourth and third Markov numbers, respectively, where the sixth member in this sequence (34) is the magic constant of a normal magic octagram and magic square.[67] In between these three Markov numbers is the tenth prime number 29[13] that represents the number of pentacubes when reflections are considered distinct; this number is also the fifth Lucas prime after 11 and 7 (where the first prime that is not a Lucas prime is 5, followed by 13).[68] A magic constant of 505 is generated by a normal magic square,[67] where 10 is the fifth composite.[69]
5 is also the value of the central cell the only non-trivial normal magic hexagon made of nineteen cells.[70][b] In a two-dimensional array, the number of planar partitions with a sum of five is twenty-four,[71] a value equal to the sum-of-divisors of the ninth arithmetic number 15[72] and perfect totient number following 3 and 9,[73] whose divisors (of 15) also produce an integer arithmetic mean of 6[74] alongside an aliquot sum of 9.[35] The smallest value that the magic constant of a five-pointed magic pentagram can have using distinct integers is 24.[75][c]
Geometric properties
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Five_Pointed_Star_Lined.svg/135px-Five_Pointed_Star_Lined.svg.png)
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, . 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.[76]
The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol {5/2}) 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).[77]
Graphs theory, and planar geometry
In graph theory, all graphs with four or fewer vertices are planar, however, there is a graph with five vertices that is not: K5, the complete graph with five vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram. By Kuratowski's theorem, a finite graph is planar iff it does not contain a subgraph that is a subdivision of K5, or the complete bipartite utility graph K3,3.[78] A similar graph is the Petersen graph, which is strongly connected and also nonplanar. It is most easily described as graph of a pentagram embedded inside a pentagon, with a total of 5 crossings, a girth of 5, and a Thue number of 5.[79][80] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.[81] The automorphism group of the Petersen graph is the symmetric group of order 120 = 5!. For polynomial equations of degree 4 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 is a solvable group for ⩽ , and not for ⩾ .
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.[82][83] 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.[84] The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.[85]
Polyhedral geometry
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/De_divina_proportione_-_Illustration_13%2C_crop.jpg/170px-De_divina_proportione_-_Illustration_13%2C_crop.jpg)
There are five Platonic solids in three-dimensional space that are regular: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.[86] 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:
- Regular polyhedron compounds: the compound of five tetrahedra, compound of ten tetrahedra, compound of five cubes, compound of five octahedra, and stella octangula.[87] Icosahedral symmetry is isomorphic to the alternating group on five letters of order 120, realized by actions on these uniform polyhedron compounds (aside from the regular compound of two tetrahedra). All fifteen mirror planes of pass through the edges of a regular spherical compound of five octahedra, whose sets of three orthogonal great circles use five colors.[d][88][89]
- Space-filling convex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium.[90] The cube is the only Platonic solid that can tessellate space on its own, and the truncated octahedron and gyrobifastigium are the only Archimedean and Johnson solids, respectively, that can tessellate space with their own copies.
- Cell-transitive parallelohedra: any parallelepiped, as well as the rhombic dodecahedron, the elongated dodecahedron, the hexagonal prism and the truncated octahedron.[91] The cube is a special case of a parallelepiped, and the rhombic dodecahedron (with five stellations per Miller's rules) is the only Catalan solid to tessellate space on its own.[92]
- Regular abstract polyhedra, which include the excavated dodecahedron and the dodecadodecahedron.[93] They have combinatorial symmetries transitive on flags of their elements, with topologies equivalent to that of toroids and the ability to tile the hyperbolic plane.
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.[94]
Four-dimensional space
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Schlegel_wireframe_5-cell.png/170px-Schlegel_wireframe_5-cell.png)
The pentatope, or 5-cell, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry of order 120 = 5! and 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.[95]: p.120
- A regular 120-cell, the dual polychoron to the regular 600-cell, can fit one hundred and twenty 5-cells. Also, five 24-cells fit inside a small stellated 120-cell, the first stellation of the 120-cell.
A subset of the vertices of the small stellated 120-cell are matched by the great duoantiprism star, which is the only uniform nonconvex duoantiprismatic solution in the fourth dimension, constructed from the polytope cartesian product and made of fifty tetrahedra, ten pentagrammic crossed antiprisms, ten pentagonal antiprisms, and fifty vertices.[95]: p.124 - The abstract four-dimensional 57-cell is made of fifty-seven hemi-icosahedral cells, in-which five surround each edge.[96] The 11-cell, another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven hemi-dodecahedral cells each with fifteen edges.[97] The skeleton of the hemi-dodecahedron is the Petersen graph.
Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora based on only twenty-five uniform polyhedra: , , , , and , 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 hexadecachoric or icositetrachoric symmetry do not exist in dimensions ⩾ ; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have and 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.[98] Only two regular projective polytopes exist in each higher dimensional space.
Generally, star polytopes that are regular only exist in dimensions ⩽ < , and can be constructed using five Miller rules for stellating polyhedra or higher-dimensional polytopes.[99]
Five-dimensional space
The 5-simplex or hexateron is the five-dimensional analogue of the 5-cell, or 4-simplex. It has Coxeter group as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group , 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 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.[100] There are also exclusively twelve complex aperiotopes in complex spaces of dimensions ⩾ ; alongside complex polytopes in and higher under simplex, hypercubic and orthoplex groups (with van Oss polytopes).[101]
Veronese surface
A Veronese surface in the projective plane generalizes a linear condition for a point to be contained inside a conic, where five points determine a conic.[4]
In finite simple groups
Lie groups
There are five complex exceptional Lie algebras: , , , , and . The smallest of these, 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.[102] is the largest, and holds the other four Lie algebras as subgroups, with a representation over 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.[103] This sphere packing 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.[104][105] The smallest simple isomorphism found inside finite simple Lie groups is ,[106] where here represents alternating groups and 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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/SporadicGroups.png/300px-SporadicGroups.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/EllipseSubqR.svg/13px-EllipseSubqR.svg.png)
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 multiply transitive permutation groups on objects, with ∈ {11, 12, 22, 23, 24}.[107]: p.54 In particular, , 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 elements.[108] Of precisely five different conjugacy classes of maximal subgroups of , one is the almost simple symmetric group (of order 5!), and another is , also almost simple, that functions as a point stabilizer which contains five as its largest prime factor in its group order: 24·32·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas is sharply 4-transitive, is sharply 5-transitive and is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups.[109] has the first five prime numbers as its distinct prime factors in its order of 27·32·5·7·11; all Mathieu groups are subgroups of , which under the Witt design of Steiner system emerges a construction of the extended binary Golay code that has as its automorphism group.[107]: pp.39, 47, 55 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.[107]: p.38 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 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 , is in turn the subject of the second generation of seven sporadic groups.[107]: pp.99, 125
There are five non-supersingular prime numbers — 37, 43, 53, 61, and 67 — less than 71, which is the largest of fifteen supersingular primes that divide the order of the friendly giant, itself the largest sporadic group.[110] In particular, a centralizer of an element of order 5 inside this group arises from the product between Harada–Norton sporadic group and a group of order 5.[111][112] On its own, can be represented using standard generators that further dictate a condition where .[113][114] This condition is also held by other generators that belong to the Tits group ,[115] 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, holds a 133-dimensional representation where 5 acts on a commutative yet non-associative product as a 5-modular analogue of the Griess algebra ♮,[116] which holds the friendly giant as its automorphism group.
List of basic calculations
Multiplication | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5 × x | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 | 95 | 100 |
Division | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5 ÷ x | 5 | 2.5 | 1.6 | 1.25 | 1 | 0.83 | 0.714285 | 0.625 | 0.5 | 0.5 | 0.45 | 0.416 | 0.384615 | 0.3571428 | 0.3 | |
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 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5x | 5 | 25 | 125 | 625 | 3125 | 15625 | 78125 | 390625 | 1953125 | 9765625 | 48828125 | 244140625 | 1220703125 | 6103515625 | 30517578125 | |
x5 | 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 0, and vulgar fractions with 5 or 2 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 raised to the fifth power always ends in the same digit as .
Evolution of the Arabic digit
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Evolution5glyph.png/198px-Evolution5glyph.png)
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.[117] 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 .
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Seven-segment_5.svg/27px-Seven-segment_5.svg.png)
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.[118] Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.[119] Five is the number of appendages on most starfish, which exhibit pentamerism.[120]
Computing
5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.[121]
Literature
Poetry
A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.[122]
Music
Modern musical notation uses a musical staff made of five horizontal lines.[123] A scale with five notes per octave is called a pentatonic scale.[124] A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[125] 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 (חומש, Hebrew for "fifth").[126] 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.[127]
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).[128]
Islam
The Five Pillars of Islam.[129]
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: dharti, agni, jal, vayu evam akash (earth, fire, water, air and space, respectively). In East Asian tradition, there are five elements: water, fire, earth, wood, and metal.[130] 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.[131] 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.[132]
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.[133] The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.
Miscellaneous fields
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/75/5_playing_cards.jpg/220px-5_playing_cards.jpg)
- "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).[134]
- The number of dots in a quincunx.[135]
See also
5 (disambiguation)
Notes
References
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- ^ Weisstein, Eric W. "Twin Primes". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
- ^ a b Dixon, A. C. (March 1908). "The Conic through Five Given Points". The Mathematical Gazette. 4 (70). The Mathematical Association: 228–230. doi:10.2307/3605147. JSTOR 3605147. S2CID 125356690.
- ^ Kantor, I. L.; Solodownikow, A. S. (1989). Hypercomplex Numbers: An Elementary Introduction to Algebras. Translated by Shenitzer., A. New York, NY: Springer-Verlag. pp. 109–110. ISBN 978-1-4612-8191-7. OCLC 19515061. S2CID 60314285.
- ^ Imaeda, K.; Imaeda, M. (2000). "Sedenions: algebra and analysis". Applied Mathematics and Computation. 115 (2). Amsterdam, Netherlands: Elsevier: 77–88. doi:10.1016/S0096-3003(99)00140-X. MR 1786945. S2CID 32296814. Zbl 1032.17003.
- ^ Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A006562 (Balanced primes (of order one): primes which are the average of the previous prime and the following prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes p: (p-1)/2 is also prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
- ^ Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-14.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-08.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A000668 (Mersenne primes (primes of the form 2^n - 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-03.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A103901 (Mersenne primes p such that M(p) equal to 2^p - 1 is also a (Mersenne) prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-03.
- ^ Conrad, Keith E. "Example of Mordell's Equation" (PDF) (Professor Notes). University of Connecticut (Homepage). p. 10. S2CID 5216897.
- ^ Weisstein, Eric W. "Sierpiński Number of the First Kind". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ Sloane, N. J. A. (ed.). "Sequence A019434 (Fermat primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21.
- ^ Sloane, N. J. A. (ed.). "Sequence A004729 (... the 31 regular polygons with an odd number of sides constructible with ruler and compass)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26.
- ^ a b Conway, John H.; Guy, Richard K. (1996). The Book of Numbers. New York, NY: Copernicus (Springer). pp. ix, 1–310. doi:10.1007/978-1-4612-4072-3. ISBN 978-1-4612-8488-8. OCLC 32854557. S2CID 115239655.
- ^ Sloane, N. J. A. (ed.). "Sequence A000127 (Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-31.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A007540 (Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A080076 (Proth primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
- ^ Sloane, N. J. A. (ed.). "Sequence A088054 (Factorial primes: primes which are within 1 of a factorial number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A005165 (Alternating factorials: n! - (n-1)! + (n-2)! - ... 1!)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-09.
- ^ Bourcereau (2015-08-19). "28". Prime Curios!. PrimePages. Retrieved 2022-10-13.
The only known number which can be expressed as the sum of the first non-negative integers (1 + 2 + 3 + 4 + 5 + 6 + 7), the first primes (2 + 3 + 5 + 7 + 11) and the first non-primes (1 + 4 + 6 + 8 + 9). There is probably no other number with this property.
- ^ Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers: numbers n such that the harmonic mean of the divisors of n is an integer.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A001600 (Harmonic means of divisors of harmonic numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A019279 (Superperfect numbers: numbers k such that sigma(sigma(k)) equals 2*k where sigma is the sum-of-divisors function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-26.
- ^ Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 84–86. ISBN 0-387-20860-7.
- ^ Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
- ^ Sloane, N. J. A. (ed.). "Sequence A007691 (Multiply-perfect numbers: n divides sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-28.
- ^ a b c Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-11.
- ^ Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
- ^ Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
- ^ Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
- ^ Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers...Sum of two squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
- ^ Sloane, N. J. A. (ed.). "Sequence A103606 (Primitive Pythagorean triples in nondecreasing order of perimeter, with each triple in increasing order, and if perimeters coincide then increasing order of the even members.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
- ^ Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
- ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08. In general, the sum of n consecutive triangular numbers is the nth tetrahedral number.
- ^ Sloane, N. J. A. (ed.). "Sequence A000332 (Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-14.
- ^ Pomerance, Carl; Yang, Hee-Sung (14 June 2012). "On Untouchable Numbers and Related Problems" (PDF). math.dartmouth.edu. Dartmouth College: 1. S2CID 30344483. 2010 Mathematics Subject Classification. 11A25, 11Y70, 11Y16.
- ^ Helfgott, Harald Andres (2014). "The ternary Goldbach problem" (PDF). In Jang, Sun Young (ed.). Seoul International Congress of Mathematicians Proceedings. Vol. 2. Seoul, KOR: Kyung Moon SA. pp. 391–418. ISBN 978-89-6105-805-6. OCLC 913564239.
- ^ Tao, Terence (March 2014). "Every odd number greater than 1 has a representation is the sum of at most five primes" (PDF). Mathematics of Computation. 83 (286): 997–1038. doi:10.1090/S0025-5718-2013-02733-0. MR 3143702. S2CID 2618958.
- ^ Sellers, James A. (2013). "An unexpected congruence modulo 5 for 4-colored generalized Frobenius partitions". J. Indian Math. Soc. New Series (Special Issue). Pune, IMD: Indian Mathematical Society: 99. arXiv:1302.5708. Bibcode:2013arXiv1302.5708S. MR 0157339. S2CID 116931082. Zbl 1290.05015.
- ^ Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1980). An Introduction to the Theory of Numbers (5th ed.). New York, NY: John Wiley. pp. 144, 145. ISBN 978-0-19-853171-5.
- ^ Sloane, N. J. A. (ed.). "Sequence A047701 (All positive numbers that are not the sum of 5 nonzero squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-20.
- Only twelve integers up to 33 cannot be expressed as the sum of five non-zero squares: {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33} where 2, 3 and 7 are the only such primes without an expression.
- ^ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem: g(n) equal to 2^n + floor((3/2)^n) - 2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-20.
- ^ Hilbert, David (1909). "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)". Mathematische Annalen (in German). 67 (3): 281–300. doi:10.1007/bf01450405. JFM 40.0236.03. MR 1511530. S2CID 179177986.
- ^ Weisstein, Eric W. "Perrin Sequence". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ M.J. Bertin; A. Decomps-Guilloux; M. Grandet-Hugot; M. Pathiaux-Delefosse; J.P. Schreiber (1992). Pisot and Salem Numbers. Birkhäuser. ISBN 3-7643-2648-4.
- ^ Sloane, N. J. A. (ed.). "Sequence A000931 (Padovan sequence (or Padovan numbers): a(n) is a(n-2) + a(n-3) with a(0) equal to 1, a(1) equal to a(2) equal to 0.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-09.
- ^ Sloane, N. J. A. (ed.). "Sequence A001609". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A000930 (Narayana's cows sequence: a(0) is a(1) equal to a(2) equal to 1; thereafter a(n) is a(n-1) + a(n-3).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-09.
- ^ Sloane, N. J. A. (ed.). "Sequence A000129 (Pell numbers: a(0) is 0, a(1) equal to 1; for n greater than 1, a(n) is 2*a(n-1) + a(n-2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-09.
- ^ a b c Böttcher, Julia; Foniok, Jan (2013). "Ramsey Properties of Permutations". The Electronic Journal of Combinatorics. 20 (1): P2. arXiv:1103.5686v2. doi:10.37236/2978. S2CID 17184541. Zbl 1267.05284.
- ^ William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 2022-07-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A006881 (Squarefree semiprimes: Numbers that are the product of two distinct primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A003101 (a(n) as Sum_{k equal to 1..n} (n - k + 1)^k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A000567 (Octagonal numbers: n*(3*n-2). Also called star numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ "Sloane's A008277 :Triangle of Stirling numbers of the second kind". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-12-24.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
- ^ Sloane, N. J. A. (ed.). "Sequence A000162 (Number of 3-dimensional polyominoes (or polycubes) with n cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
- ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-25.
- ^ Trigg, C. W. (February 1964). "A Unique Magic Hexagon". Recreational Mathematics Magazine. Retrieved 2022-07-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-10.
- ^ Sloane, N. J. A. (ed.). "Sequence A000203 (a(n) is sigma(n), the sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-10.
- ^ Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-14.
- ^ Gardner, Martin (1989). Mathematical Carnival. Mathematical Games (5th ed.). Washington, D.C.: Mathematical Association of America. pp. 56–58. ISBN 978-0-88385-448-8. OCLC 20003033. Zbl 0684.00001.
- ^ Sloane, N. J. A. (ed.). "Sequence A307681 (Difference between the number of sides and the number of diagonals of a convex n-gon.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sarhangi, Reza (2012). "Interlocking Star Polygons in Persian Architecture: The Special Case of the Decagram in Mosaic Designs" (PDF). Nexus Network Journal. 14 (2): 350. doi:10.1007/s00004-012-0117-5. S2CID 124558613.
- ^ Burnstein, Michael (1978). "Kuratowski-Pontrjagin theorem on planar graphs". Journal of Combinatorial Theory. Series B. 24 (2): 228–232. doi:10.1016/0095-8956(78)90024-2.
- ^ Holton, D. A.; Sheehan, J. (1993). The Petersen Graph. Cambridge University Press. pp. 9.2, 9.5 and 9.9. ISBN 0-521-43594-3.
- ^ Alon, Noga; Grytczuk, Jaroslaw; Hałuszczak, Mariusz; Riordan, Oliver (2002). "Nonrepetitive colorings of graphs" (PDF). Random Structures & Algorithms. 2 (3–4): 337. doi:10.1002/rsa.10057. MR 1945373. S2CID 5724512.
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.
- ^ Royle, G.; et al. (February 2001). "Cubic Symmetric Graphs (The Foster Census)". Archived from the original on 2008-07-20.
- ^ de Grey, Aubrey D.N.J. (2018). "The Chromatic Number of the Plane is At Least 5". Geombinatorics. 28: 5–18. arXiv:1804.02385. MR 3820926. S2CID 119273214.
- ^ Exoo, Geoffrey; Ismailescu, Dan (2020). "The Chromatic Number of the Plane is At Least 5: A New Proof". Discrete & Computational Geometry. 64. New York, NY: Springer: 216–226. arXiv:1805.00157. doi:10.1007/s00454-019-00058-1. MR 4110534. S2CID 119266055. Zbl 1445.05040.
- ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. 50 (5). Taylor & Francis, Ltd.: 227–236. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
- ^ Grünbaum, Branko; Shephard, Geoffrey C. (1987). "Tilings by polygons". Tilings and Patterns. New York: W. H. Freeman and Company. ISBN 978-0-7167-1193-3. MR 0857454. Section 9.3: "Other Monohedral tilings by convex polygons".
- ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61
- ^ Skilling, John (1976). "Uniform Compounds of Uniform Polyhedra". Mathematical Proceedings of the Cambridge Philosophical Society. 79 (3): 447–457. Bibcode:1976MPCPS..79..447S. doi:10.1017/S0305004100052440. MR 0397554. S2CID 123279687.
- ^ Hart, George W. (1998). "Icosahedral Constructions" (PDF). In Sarhangi, Reza (ed.). Bridges: Mathematical Connections in Art, Music, and Science. Winfield, Kansas: The Bridges Organization. p. 196. ISBN 978-0-9665201-0-1. OCLC 59580549. S2CID 202679388.
- ^ Hart, George W. "Symmetry Planes". Virtual Polyhedra (The Encyclopedia of Polyhedra). Retrieved 2023-09-27.
- "They can be colored as five sets of three mutually orthogonal planes" where the "fifteen planes divide the sphere into 120 Möbius triangles."
- ^ Kepler, Johannes (2010). The Six-Cornered Snowflake. Paul Dry Books. Footnote 18, p. 146. ISBN 978-1-58988-285-0.
- ^ Alexandrov, A. D. (2005). "8.1 Parallelohedra". Convex Polyhedra. Springer. pp. 349–359.
- ^ Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 2022-11-26. Retrieved 2023-01-12.
- ^ Wills, J. M. (1987). "The combinatorially regular polyhedra of index 2". Aequationes Mathematicae. 34 (2–3): 206–220. doi:10.1007/BF01830672. S2CID 121281276.
- ^ Har'El, Zvi (1993). "Uniform Solution for Uniform Polyhedra" (PDF). Geometriae Dedicata. 47. Netherlands: Springer Publishing: 57–110. doi:10.1007/BF01263494. MR 1230107. S2CID 120995279. Zbl 0784.51020.
- "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.
- "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."
- ^ a b H. S. M. Coxeter (1973). Regular Polytopes (3rd ed.). New York: Dover Publications, Inc. pp. 1–368. ISBN 978-0-486-61480-9.
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- ^ Coxeter, H. S. M (1984). "A Symmetrical Arrangement of Eleven Hemi-Icosahedra". Annals of Discrete Mathematics. North-Holland Mathematics Studies. 87 (20): 103–114. doi:10.1016/S0304-0208(08)72814-7. ISBN 978-0-444-86571-7.
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There are five basic tastes: sweet, salty, sour, bitter and umami...
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The typical limb of tetrapods is the pentadactyl limb (Gr. penta, five) that has five toes. Tetrapods evolved from an ancestor that had limbs with five toes. ... Even though the number of digits in different vertebrates may vary from five, vertebrates develop from an embryonic five-digit stage.
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The five appendages of the starfish are thought to be homologous to five human buds
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5 5 005 ENQ (enquiry)
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The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter)
- ^ "STAVE | meaning in the Cambridge English Dictionary". dictionary.cambridge.org. Retrieved 2020-08-02.
the five lines and four spaces between them on which musical notes are written
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Pentatonic scales, as used in jazz, are five note scales
- ^ Danneley, John Feltham (1825). 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. editor, and pub.
are the perfect fourth, perfect fifth, and the octave
- ^ Pelaia, Ariela. "Judaism 101: What Are the Five Books of Moses?". Learn Religions. Retrieved 2020-08-03.
- ^ Zenner, Walter P. (1988-01-01). Persistence and Flexibility: Anthropological Perspectives on the American Jewish Experience. SUNY Press. p. 284. ISBN 978-0-88706-748-8.
- ^ "CATHOLIC ENCYCLOPEDIA: The Five Sacred Wounds". www.newadvent.org. Retrieved 2020-08-02.
- ^ "PBS – Islam: Empire of Faith – Faith – Five Pillars". www.pbs.org. Retrieved 2020-08-03.
- ^ Yoon, Hong-key (2006). The Culture of Fengshui in Korea: An Exploration of East Asian Geomancy. Lexington Books. p. 59. ISBN 978-0-7391-1348-6.
The first category is the Five Agents [Elements] namely, Water, Fire, Wood, Metal, and Earth.
- ^ Walsh, Len (2008-11-15). Read Japanese Today: The Easy Way to Learn 400 Practical Kanji. Tuttle Publishing. ISBN 978-1-4629-1592-7.
The Japanese names of the days of the week are taken from the names of the seven basic nature symbols
- ^ Chen, Yuan (2014). "Legitimation Discourse and the Theory of the Five Elements in Imperial China". Journal of Song-Yuan Studies. 44 (1): 325–364. doi:10.1353/sys.2014.0000. ISSN 2154-6665. S2CID 147099574.
- ^ Kronland-Martinet, Richard; Ystad, Sølvi; Jensen, Kristoffer (2008-07-19). Computer Music Modeling and Retrieval. Sense of Sounds: 4th International Symposium, CMMR 2007, Copenhagen, Denmark, August 2007, Revised Papers. Springer. p. 502. ISBN 978-3-540-85035-9.
Plato and Aristotle postulated a fifth state of matter, which they called "idea" or quintessence" (from "quint" which means "fifth")
- ^ "Olympic Rings – Symbol of the Olympic Movement". International Olympic Committee. 2020-06-23. Retrieved 2020-08-02.
- ^ Laplante, Philip A. (2018-10-03). Comprehensive Dictionary of Electrical Engineering. CRC Press. p. 562. ISBN 978-1-4200-3780-7.
quincunx five points
Further reading
- Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. London, UK: Penguin Group. pp. 58–67.
External links
- Prime curiosities: 5
Media related to 5 (number) at Wikimedia Commons
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