5

5 is the fifth Fibonacci number, being 2 plus 3,[2] and the only Fibonacci number that is equal to its position aside from 1 (that is also the second index). Five is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation: (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). In the Perrin sequence 5 is both the fifth and sixth Perrin numbers.[18]

5 is the second Fermat prime of the form ${\displaystyle 2^{2^{n}}+1}$, and more generally the second Sierpiński number of the first kind, ${\displaystyle n^{n}+1}$.[19] There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.[20] The sum of the first three Fermat primes, 3, 5 and 17, yields 25 or 5², while 257 is the 55th prime number. 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.[21][22]^{: 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 ${\displaystyle n}$-sided polygons, which is equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem.[23][22]^{: pp.76–78}

5 is also the third Mersenne prime exponent of the form ${\displaystyle 2^{n}-1}$, which yields ${\displaystyle 31}$, the eleventh prime number and fifth super-prime.[24][2] This is the prime index of the third Mersenne prime and second double Mersenne prime 127,[25] as well as the third double Mersenne prime exponent for the number 2,147,483,647,[25] which is the largest value that a signed 32-bit integer field can hold. There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime ${\displaystyle M_{M_{61}}}$ = 2^{23058...93951} − 1 too large to compute with current computers. In a related sequence, the first five terms in the sequence of Catalan–Mersenne numbers ${\displaystyle M_{c_{n}}}$ are the only known prime terms, with a sixth possible candidate in the order of 10^1037.7094. These prime sequences are conjectured to be prime up to a certain limit.

There are a total of five known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors.[26][27] The smallest such number is 6, and the largest of these is equivalent to the sum of 4095 divisors, where 4095 is the largest of five Ramanujan–Nagell numbers that are both triangular numbers and Mersenne numbers of the general form.[28][29] 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 ${\displaystyle 2^{p-1}}$(${\displaystyle 2^{p}-1}$) with a ${\displaystyle p}$ of ${\displaystyle 5}$, by the Euclid–Euler theorem.[30][31][32] 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.[33][34] The fifth Mersenne prime, 8191,[24] splits into 4095 and 4096, with the latter being the fifth superperfect number[35] and the sixth power of four, 4⁶.

In figurate numbers, 5 is a pentagonal number, with the sequence of pentagonal numbers starting: 1, 5, 12, 22, 35, ...[36]

The factorial of five ${\displaystyle 5!=120}$ is multiply perfect like 28 and 496.[41] 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, ${\displaystyle 120+5=125=5^{3}}$, where 125 is the second number to have an aliquot sum of 31 (after the fifth power of two, 32).[42] On its own, 31 is the first prime centered pentagonal number,[43] and the fifth centered triangular number.[44] Collectively, five and thirty-one generate a sum of 36 (the square of 6) and a difference of 26, which is the only number to lie between a square ${\displaystyle a^{2}}$ and a cube ${\displaystyle b^{3}}$ (respectively, 25 and 27).[45] 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.[46] 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, ...[47] The first five members in this sequence add to 126, which is also the sixth pentagonal pyramidal number[48] as well as the fifth ${\displaystyle {\mathcal {S}}}$-perfect Granville number.[49] This is the third Granville number not to be perfect, and the only known such number with three distinct prime factors.[50]

55 is the fifteenth discrete biprime,[51] equal to the product between 5 and the fifth prime and third super-prime 11.[2] These two numbers also form the second pair (5, 11) of Brown numbers ${\displaystyle (n,m)}$ such that ${\displaystyle n!+1=m^{2}}$ where five is also the second number that belongs to the first pair (4, 5); altogether only five distinct numbers (4, 5, 7, 11, and 71) are needed to generate the set of known pairs of Brown numbers, where the third and largest pair is (7, 71).[52][53] Fifty-five is also the tenth Fibonacci number,[54] whose digit sum is also 10, in its decimal representation. It is the tenth triangular number and the fourth that is doubly triangular,[55] the fifth heptagonal number[56] and fourth centered nonagonal number,[57] and as listed above, the fifth square pyramidal number.[38] The sequence of triangular ${\displaystyle n}$ that are powers of 10 is: 55, 5050, 500500, ...[58] 55 in base-ten is also the fourth Kaprekar number as are all triangular numbers that are powers of ten, which initially includes 1, 9 and 45,[59] with forty-five itself the ninth triangular number where 5 lies midway between 1 and 9 in the sequence of natural numbers. 45 is also conjectured by Ramsey number ${\displaystyle R(5,5)}$,[60][61] and is a Schröder–Hipparchus number; the next and fifth such number is 197, the forty-fifth prime number that represents the number of ways of dissecting a heptagon into smaller polygons by inserting diagonals.[62] A five-sided convex pentagon, on the other hand, has eleven ways of being subdivided in such manner.[lower-alpha 1]

The smallest non-trivial magic square

5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. Its ${\displaystyle 3\times 3}$ array has a magic constant ${\displaystyle \mathrm {M} }$ of ${\displaystyle 15}$, where the sums of its rows, columns, and diagonals are all equal to fifteen.[63] On the other hand, a normal ${\displaystyle 5\times 5}$ magic square[lower-alpha 2] has a magic constant ${\displaystyle \mathrm {M} }$ of ${\displaystyle 65=13\times 5}$, where 5 and 13 are the first two Wilson primes.[4] The fifth number to return ${\displaystyle 0}$ for the Mertens function is 65,[64] with ${\displaystyle M(x)}$ counting the number of square-free integers up to ${\displaystyle x}$ 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,[51] with an aliquot sum of 19 as well[42] and equivalent to 1⁵ + 2⁴ + 3³ + 4² + 5¹.[65] It is also the magic constant of the ${\displaystyle n-}$Queens Problem for ${\displaystyle n=5}$,[66] the fifth octagonal number,[67] and the Stirling number of the second kind ${\displaystyle S(6,4)}$ that represents sixty-five ways of dividing a set of six objects into four non-empty subsets.[68] 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 ${\displaystyle 4\times 4}$ magic square.[69] In between these three Markov numbers is the tenth prime number 29 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).[70] A magic constant of 505 is generated by a ${\displaystyle 10\times 10}$ normal magic square,[69] where 10 is the fifth composite.[71]

5 is also the value of the central cell the only non-trivial normal magic hexagon made of nineteen cells.[72][lower-alpha 3] Where the sum between the magic constants of this order-3 normal magic hexagon (38) and the order-5 normal magic square (65) is 103 — the prime index of the third Wilson prime 563 equal to the sum of all three pairs of Brown numbers — their difference is 27, itself the prime index of 103.[73] In base-ten, 15 and 27 are the only two-digit numbers that are equal to the sum between their digits (inclusive, i.e. 2 + 3 + ... + 7 = 27), with these two numbers consecutive perfect totient numbers after 3 and 9.[74] 103 is the fifth irregular prime[75] that divides the numerator (236364091) of the twenty-fourth Bernoulli number ${\displaystyle B_{24}}$, and as such it is part of the eighth irregular pair (103, 24).[76] In a two-dimensional array, the number of planar partitions with a sum of four is equal to thirteen and the number of such partitions with a sum of five is twenty-four,[77] a value equal to the sum-of-divisors of the ninth arithmetic number 15[78] whose divisors also produce an integer arithmetic mean of 6[79] (alongside an aliquot sum of 9).[42] The smallest value that the magic constant of a five-pointed magic pentagram can have using distinct integers is 24.[80][lower-alpha 4]

In the Collatz 3x + 1 problem, 5 requires five steps to reach one by multiplying terms by three and adding one if the term is odd (starting with five itself), and dividing by two if they are even: {5 ➙ 16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}; the only other number to require five steps is 32 since 16 must be part of such path (see [lower-alpha 5] for a map of orbits for small odd numbers).[81][82]

Specifically, 120 needs fifteen steps in total to arrive at 5: {120 ➙ 60 ➙ 30 ➙ 15 ➙ 46 ➙ 23 ➙ 70 ➙ 35 ➙ 106 ➙ 53 ➙ 160 ➙ 80 ➙ 40 ➙ 20 ➙ 10 ➙ 5}. These comprise a total of sixteen numbers before cycling through {16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}. On the other hand, the trajectory of 15 requires seventeen steps to reach 1,[82] where its reduced Collatz trajectory is equal to five when counting the steps {23, 35, 53, 5, 1} that are prime, including 1.[83] Overall, thirteen numbers in the Collatz map for 15 are composite,[81] where the largest prime in its trajectory 53 is the sixteenth prime number.

When generalizing the Collatz conjecture to all positive or negative integers, −5 becomes one of only four known possible cycle starting points and endpoints, and in its case in five steps too: {−5 ➙ −14 ➙ −7 ➙ −20 ➙ −10 ➙ −5 ➙ ...}. The other possible cycles begin and end at −17 in eighteen steps, −1 in two steps, and 1 in three steps. This behavior is analogous to the path cycle of five in the 3x − 1 problem, where 5 takes five steps to return cyclically, in this instance by multiplying terms by three and subtracting 1 if the terms are odd, and also halving if even.[84] It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).[85]

Five is conjectured to be the only odd untouchable number, and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.[86] Meanwhile:

While all integers ${\displaystyle n\geq 34}$ can be expressed as the sum of five non-zero squares,[89][90] ${\displaystyle g(5)=37}$ in Waring's problem, where every natural number ${\displaystyle n\in \mathbb {N} }$ is the sum of at most thirty-seven fifth powers.[91][92]

Polynomial equations of degree 4 and below can be solved with radicals, while 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 ${\displaystyle \mathrm {S} _{n}}$ is a solvable group for ${\displaystyle n}$ ⩽ ${\displaystyle 4}$, and not for ${\displaystyle n}$ ⩾ ${\displaystyle 5}$.

There are five countably infinite Ramsey classes of permutations, where the age of each countable homogeneous permutation forms an individual Ramsey class ${\displaystyle K}$ of objects such that, for each natural number ${\displaystyle r}$ and each choice of objects ${\displaystyle A,B\in K}$, there is no object ${\displaystyle C\in K}$ where in any ${\displaystyle r}$-coloring of all subobjects of ${\displaystyle C}$ isomorphic to ${\displaystyle A}$ there exists a monochromatic subobject isomorphic to ${\displaystyle B}$.[93]^{: pp.1, 2} Aside from ${\displaystyle \{1\}}$, the five classes of Ramsey permutations are the classes of:[93]^: p.4

In general, the Fraïssé limit of a class ${\displaystyle K}$ of finite relational structure is the age of a countable homogeneous relational structure ${\displaystyle U}$ if and only if five conditions hold for ${\displaystyle K}$: 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.[93]^: p.3

In the general classification of number systems, the real numbers ${\displaystyle \mathbb {R} }$ and its three subsequent Cayley-Dickson constructions of algebras over the field of the real numbers (i.e. the complex numbers ${\displaystyle \mathbb {C} }$, the quaternions ${\displaystyle \mathbb {H} }$, and the octonions ${\displaystyle \mathbb {O} }$) 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.[94] Whereas the real numbers contain all five properties, the octonions are only alternative and power-associative. On the other hand, the sedenions ${\displaystyle \mathbb {S} }$, which represent a fifth algebra in this series, is not a composition algebra unlike ${\displaystyle \mathbb {H} }$ and ${\displaystyle \mathbb {O} }$, is only power-associative, and is the first algebra to contain non-trivial zero divisors as with all further algebras over larger fields.[95] Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16.

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, ${\displaystyle \varphi }$. Its internal geometry appears prominently in Penrose tilings, and is a facet inside Kepler-Poinsot star polyhedra and Schläfli–Hess star polychora, represented by its Schläfli symbol {5/2}. A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges. It is often found as a facet inside Islamic Girih tiles, of which there are five different rudimentary types.[96] Generally, star polytopes that are regular only exist in dimensions ${\displaystyle 2}$ ⩽ ${\displaystyle n}$ < ${\displaystyle 5}$, and can be constructed using five Miller rules for stellating polyhedra or higher-dimensional polytopes.[97]

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 K_3,3.[98] 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.[99][100] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.[101] The automorphism group of the Petersen graph is the symmetric group ${\displaystyle \mathrm {S} _{5}}$ of order 120 = 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.[102][103] 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.[104] The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.[105]

Illustration by Leonardo da Vinci of a regular dodecahedron, from Luca Pacioli's Divina proportione

There are five Platonic solids in three-dimensional space that are regular: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.[106] 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, the fifth pentagonal pyramidal number ${\displaystyle 75=15\times 5}$ represents the total number of indexed uniform compound polyhedra,[107] which includes seven families of prisms and antiprisms. Seventy-five is also the number of non-prismatic uniform polyhedra, which includes Platonic solids, Archimedean solids, and star polyhedra; 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-antirprism.[114] In all, there are twenty-five uniform polyhedra that generate four-dimensional uniform polychora, they are the five Platonic solids, fifteen Archimedean solids counting two enantiomorphic forms, and five associated prisms: the triangular, pentagonal, hexagonal, octagonal, and decagonal prisms.

The four-dimensional 5-cell is the simplest regular polychoron.

The pentatope, or 5-cell, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry ${\displaystyle \mathrm {A} _{4}}$ of order 120 = 5! and ${\displaystyle \mathrm {S} _{5}}$ 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.[115]^: p.120

Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora based on only twenty-five uniform polyhedra: ${\displaystyle \mathrm {A} _{4}}$, ${\displaystyle \mathrm {B} _{4}}$, ${\displaystyle \mathrm {D} _{4}}$, ${\displaystyle \mathrm {F} _{4}}$, and ${\displaystyle \mathrm {H} _{4}}$, 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 ${\displaystyle \mathrm {H} _{4}}$ hexadecachoric or ${\displaystyle \mathrm {F} _{4}}$ icositetrachoric symmetry do not exist in dimensions ${\displaystyle n}$ ⩾ ${\displaystyle 5}$; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have ${\displaystyle \mathrm {H} _{4}}$ and ${\displaystyle \mathrm {F} _{4}}$ 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.[119] Only two regular projective polytopes exist in each higher dimensional space.

The fundamental polygon for Bring's curve is a regular hyperbolic twenty-sided icosagon.

In particular, Bring's surface is the curve in the projective plane ${\displaystyle \mathbb {P} ^{4}}$ that is represented by the homogeneous equations:[120]

${\displaystyle v+w+x+y+z=v^{2}+w^{2}+x^{2}+y^{2}+z^{2}=v^{3}+w^{3}+x^{3}+y^{3}+z^{3}=0.}$

It holds the largest possible automorphism group of a genus four complex curve, with group structure ${\displaystyle \mathrm {S} _{5}}$. This is the Riemann surface associated with the small stellated dodecahedron, whose fundamental polygon is a regular hyperbolic icosagon, with an area of ${\displaystyle 12\pi }$ (by the Gauss-Bonnet theorem). Including reflections, its full group of symmetries is ${\displaystyle \mathrm {S} _{5}\times \mathbb {Z} _{2}}$, of order 240; which is also the number of (2,4,5) hyperbolic triangles that tessellate its fundamental polygon. Bring quintic ${\displaystyle x^{5}+ax+b=0}$ holds roots ${\displaystyle x_{i}}$ that satisfy Bring's curve.

The 5-simplex or hexateron is the five-dimensional analogue of the 5-cell, or 4-simplex. It has Coxeter group ${\displaystyle \mathrm {A} _{5}}$ as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group ${\displaystyle \mathrm {S} _{6}}$, 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 ${\displaystyle \mathrm {B} _{5}}$ 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.[121] There are also exclusively twelve complex aperiotopes in ${\displaystyle \mathbb {C} ^{n}}$ complex spaces of dimensions ${\displaystyle n}$ ⩾ ${\displaystyle 5}$; alongside complex polytopes in ${\displaystyle \mathbb {C} ^{5}}$ and higher under simplex, hypercubic and orthoplex groups (with van Oss polytopes).[122]

A Veronese surface in the projective plane ${\displaystyle \mathbb {P} ^{5}}$ generalizes a linear condition ${\displaystyle \nu$ :\mathbb {P} ^{2}\to \mathbb {P} ^{5}} for a point to be contained inside a conic, which requires five points in the same way that two points are needed to determine a line.[123]

There are five complex exceptional Lie algebras: ${\displaystyle {\mathfrak {g}}_{2}}$, ${\displaystyle {\mathfrak {f}}_{4}}$, ${\displaystyle {\mathfrak {e}}_{6}}$, ${\displaystyle {\mathfrak {e}}_{7}}$, and ${\displaystyle {\mathfrak {e}}_{8}}$. The smallest of these, ${\displaystyle {\mathfrak {g}}_{2}}$ 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.[124] ${\displaystyle {\mathfrak {e}}_{8}}$ is the largest, and holds the other four Lie algebras as subgroups, with a representation over ${\displaystyle \mathbb {R} }$ 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.[125] This sphere packing ${\displaystyle \mathrm {E} _{8}}$ 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.[126][127] While there are specifically five solvable groups that are excluded from finite simple groups of Lie type, the smallest duplicate found inside finite simple Lie groups is ${\displaystyle \mathrm {A_{5}} \cong A_{1}(4)\cong A_{1}(5)}$, where here ${\displaystyle \mathrm {A_{n}} }$ represents alternating groups and ${\displaystyle A_{n}(q)}$ classical Chevalley groups. The smallest non-solvable group is the alternating group on five letters, which is also the smallest simple, non-abelian group.

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 ${\displaystyle \mathrm {M} _{n}}$ multiply transitive permutation groups on ${\displaystyle n}$ objects, with ${\displaystyle n}$ ∈ {11, 12, 22, 23, 24}.[128]^: p.54 In particular, ${\displaystyle \mathrm {M} _{11}}$, 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 ${\displaystyle n}$ elements.[129] Of precisely five different conjugacy classes of maximal subgroups of ${\displaystyle \mathrm {M} _{11}}$, one is the almost simple symmetric group ${\displaystyle \mathrm {S} _{5}}$ (of order 5!), and another is ${\displaystyle \mathrm {M} _{10}}$, 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 ${\displaystyle \mathrm {M} _{11}}$ is sharply 4-transitive, ${\displaystyle \mathrm {M} _{12}}$ is sharply 5-transitive and ${\displaystyle \mathrm {M} _{24}}$ is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups.[130] ${\displaystyle \mathrm {M} _{22}}$ has the first five prime numbers as its distinct prime factors in its order of 2⁷·3²·5·7·11, and is the smallest of five sporadic groups with five distinct prime factors in their order.[128]^: p.17 All Mathieu groups are subgroups of ${\displaystyle \mathrm {M} _{24}}$, which under the Witt design ${\displaystyle \mathrm {W} _{24}}$ of Steiner system ${\displaystyle \operatorname {S(5,8,24)} }$ emerges a construction of the extended binary Golay code ${\displaystyle \mathrm {B} _{24}}$ that has ${\displaystyle \mathrm {M} _{24}}$ as its automorphism group.[128]^{: pp.39, 47, 55} ${\displaystyle \mathrm {W} _{24}}$ 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.[128]^: 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 ${\displaystyle (0,1,2,3,\dots ,24;70)}$ 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 ${\displaystyle \mathrm {Co} _{0}}$, is in turn the subject of the second generation of seven sporadic groups.[128]^{: 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.[131] In particular, a centralizer of an element of order 5 inside this group arises from the product between Harada–Norton sporadic group ${\displaystyle \mathrm {HN} }$ and a group of order 5.[132][133] On its own, ${\displaystyle \mathrm {HN} }$ can be represented using standard generators ${\displaystyle (a,b,ab)}$ that further dictate a condition where ${\displaystyle o([a,b])=5}$.[134][135] This condition is also held by other generators that belong to the Tits group ${\displaystyle \mathrm {T} }$,[136] 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, ${\displaystyle \mathrm {HN} }$ holds a 133-dimensional representation where 5 acts on a commutative yet non-associative product as a 5-modular analogue of the Griess algebra ${\displaystyle V_{2}}$^♮,[137] which holds the friendly giant as its automorphism group.

Euler's identity, ${\displaystyle e^{i\pi }}$+ ${\displaystyle 1}$ = ${\displaystyle 0}$, contains five essential numbers used widely in mathematics: Archimedes' constant ${\displaystyle \pi }$, Euler's number ${\displaystyle e}$, the imaginary number ${\displaystyle i}$, unity ${\displaystyle 1}$, and zero ${\displaystyle 0}$.[138][139][140]

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 ${\displaystyle n}$ raised to the fifth power always ends in the same digit as ${\displaystyle n}$.

• There are five Lagrangian points in a two-body system.
• There are currently five dwarf planets in the Solar System: Ceres, Pluto, Haumea, Makemake, and Eris.[145]
• The New General Catalogue object NGC 5, a magnitude 13 spiral galaxy in the constellation Andromeda.[146]
• Messier object M5, a magnitude 7.0 globular cluster in the constellation Serpens.[147]

• There are usually considered to be five senses (in general terms).
• The five basic tastes are sweet, salty, sour, bitter, and umami.[148]
• Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.[149]

• 5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.[150]

• The god Shiva has five faces[151] and his mantra is also called panchakshari (five-worded) mantra.
• The goddess Saraswati, goddess of knowledge and intellectual is associated with panchami or the number 5.
• There are five elements in the universe according to Hindu cosmology: dharti, agni, jal, vayu evam akash (earth, fire, water, air and space respectively).
• The most sacred tree in Hinduism has 5 leaves in every leaf stunt.
• Most of the flowers have 5 petals in them.
• The epic Mahabharata revolves around the battle between Duryodhana and his 99 other brothers and the 5 pandava princes—Dharma, Arjuna, Bhima, Nakula and Sahadeva.

• There are traditionally five wounds of Jesus Christ in Christianity: the Scourging at the Pillar, the Crowning with Thorns, the wounds in Christ's hands, the wounds in Christ's feet, and the Side Wound of Christ.[152]

• The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.
• Five Seals in Sethianism
• Five Trees in the Gospel of Thomas

• The Five Pillars of Islam[153]
• Muslims pray to Allah five times a day[154]
• According to Shia Muslims, the Panjetan or the Five Holy Purified Ones are the members of Muhammad's family: Muhammad, Ali, Fatimah, Hasan, and Husayn and are often symbolically represented by an image of the Khamsa.[155]

• The Torah contains five books—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy—which 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").[156]
• The book of Psalms is arranged into five books, paralleling the Five Books of Moses.[157]
• 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.[158]

• The five sacred Sikh symbols prescribed by Guru Gobind Singh are commonly known as panj kakars or the "Five Ks" because they start with letter K representing kakka (ਕ) in the Punjabi language's Gurmukhi script. They are: kesh (unshorn hair), kangha (the comb), kara (the steel bracelet), kachhehra (the soldier's shorts), and kirpan (the sword) (in Gurmukhi: ਕੇਸ, ਕੰਘਾ, ਕੜਾ, ਕਛਹਰਾ, ਕਿਰਪਾਨ).[159] Also, there are five deadly evils: kam (lust), krodh (anger), moh (attachment), lobh (greed), and ankhar (ego).

• 5 Elements[160]
• 5 Emperors[161]

• 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.
• The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.
• In Cantonese, "five" sounds like the word "not" (character: 唔). When five appears in front of a lucky number, e.g. "58", the result is considered unlucky.
• In East Asian tradition, there are five elements: (water, fire, earth, wood, and metal).[162] 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.[163] 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.
• In numerology, 5 or a series of 555, is often associated with change, evolution, love and abundance.
• Members of The Nation of Gods and Earths, a primarily African American religious organization, call themselves the "Five-Percenters" because they believe that only 5% of mankind is truly enlightened.[164]

• James the Red Engine, a fictional character numbered 5.[165]
• Johnny 5 is the lead character in the film Short Circuit (1986)[166]
• Number Five is a character in Lorien Legacies[167]
• Numbuh 5, real name Abigail Lincoln, from Codename: Kids Next Door
• Sankara Stones, five magical rocks in Indiana Jones and the Temple of Doom that are sought by the Thuggees for evil purposes[168]
• The Mach Five Mahha-gō? (マッハ号), the racing car Speed Racer (Go Mifune in the Japanese version) drives in the anime series of the same name (known as "Mach Go! Go! Go!" in Japan)
• In the Yu-Gi-Oh! series, "The Big Five" are a group of five villainous corporate executives who work for KaibaCorp (Gansley, Crump, Johnson, Nezbitt and Leichter).
• In the works of J. R. R. Tolkien, five wizards (Saruman, Gandalf, Radagast, Alatar and Pallando) are sent to Middle-earth to aid against the threat of the Dark Lord Sauron[169]
• In the A Song of Ice and Fire series, the War of the Five Kings is fought between different claimants to the Iron Throne of Westeros, as well as to the thrones of the individual regions of Westeros (Joffrey Baratheon, Stannis Baratheon, Renly Baratheon, Robb Stark and Balon Greyjoy)[170]
• In The Wheel of Time series, the "Emond's Field Five" are a group of five of the series' main characters who all come from the village of Emond's Field (Rand al'Thor, Matrim Cauthon, Perrin Aybara, Egwene al'Vere and Nynaeve al'Meara)
• Myst uses the number 5 as a unique base counting system. In The Myst Reader series, it is further explained that the number 5 is considered a holy number in the fictional D'ni society.
• Number Five is also a character in The Umbrella Academy comic book and TV series adaptation[171]

• Towards the end of the film Monty Python and the Holy Grail (1975), the character of King Arthur repeatedly confuses the number five with the number three.
• Five Go Mad in Dorset (1982) was the first of the long-running series of The Comic Strip Presents... television comedy films[172]
• The Fifth Element (1997), a science fiction film[173]
• Fast Five (2011), the fifth installment of the Fast and Furious film series.[174]
• V for Vendetta (2005), produced by Warner Bros., directed by James McTeigue, and adapted from Alan Moore's graphic novel V for Vendetta prominently features number 5 and Roman Numeral V; the story is based on the historical event in which a group of men attempted to destroy Parliament on November 5, 1605[175]

• Modern musical notation uses a musical staff made of five horizontal lines.[176]
• A scale with five notes per octave is called a pentatonic scale.[177]
• A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[178]
• 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.
• Using the Latin root, five musicians are called a quintet.[179]
• Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.

• Five (group), a UK Boy band[180]
• The Five (composers), 19th-century Russian composers[181]
• 5 Seconds of Summer, pop band that originated in Sydney, Australia
• Five Americans, American rock band active 1965–1969[182]
• Five Finger Death Punch, American heavy metal band from Las Vegas, Nevada. Active 2005–present
• Five Man Electrical Band, Canadian rock group billed (and active) as the Five Man Electrical Band, 1969–1975[183]
• Maroon 5, American pop rock band that originated in Los Angeles, California[184]
• MC5, American punk rock band[185]
• Pentatonix, a Grammy-winning a cappella group originated in Arlington, Texas[186]
• The 5th Dimension, American pop vocal group, active 1977–present[187]
• The Dave Clark Five, a.k.a. DC5, an English pop rock group comprising Dave Clark, Lenny Davidson, Rick Huxley, Denis Payton, and Mike Smith; active 1958–1970[188]
• The Jackson 5, American pop rock group featuring various members of the Jackson family; they were billed (and active) as The Jackson 5, 1966–1975[189]
• Hi-5, Australian pop kids group, where it has several international adaptations, and several members throughout the history of the band. It was also a TV show.
• We Five: American folk rock group active 1965–1967 and 1968–1977
• Grandmaster Flash and the Furious Five: American rap group, 1970–80's[190]
• Fifth Harmony, an American girl group.[191]
• Ben Folds Five, an American alternative rock trio, 1993–2000, 2008 and 2011–2013[192]
• R5 (band), an American pop and alternative rock group, 2009–2018[193]

• The number of completed, numbered piano concertos of Ludwig van Beethoven, Sergei Prokofiev, and Camille Saint-Saëns

Stations

• Channel 5 (UK), a television channel that broadcasts in the United Kingdom[194]
• TV5 (formerly known as ABC 5) (DWET-TV channel 5 In Metro Manila) a television network in the Philippines.[195]

Series

• Babylon 5, a science fiction television series[196]
• The number 5 features in the television series Battlestar Galactica in regards to the Final Five cylons and the Temple of Five
• Hi-5 (Australian TV series), a television series from Australia[197]
• Hi-5 (British TV series), a television show from the United Kingdom
• Hi-5 Philippines a television show from the Philippines
• Odyssey 5, a 2002 science fiction television series[198]
• Tillbaka till Vintergatan, a Swedish children's television series featuring a character named "Femman" (meaning five), who can only utter the word 'five'.
• The Five (talk show): Fox News Channel roundtable current events television show, premiered 2011, so-named for its panel of five commentators.
• Yes! PreCure 5 is a 2007 anime series which follows the adventures of Nozomi and her friends. It is also followed by the 2008 sequel Yes! Pretty Cure 5 GoGo!
• The Quintessential Quintuplets is a 2019 slice of life romance anime series which follows the everyday life of five identical quintuplets and their interactions with their tutor. It has two seasons, and a final movie is scheduled in summer 2022.
• Hawaii Five-0, CBS American TV series.[199]

• The Famous Five is a series of children's books by British writer Enid Blyton
• The Power of Five is a series of children's books by British writer and screenwriter Anthony Horowitz
• The Fall of Five is a book written under the collective pseudonym Pittacus Lore in the series Lorien Legacies
• The Book of Five Rings is a text on kenjutsu and the martial arts in general, written by the swordsman Miyamoto Musashi circa 1645
• Slaughterhouse-Five is a book by Kurt Vonnegut about World War II[200]

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