Figurate numbers

In figurate numbers, 5 is a pentagonal number, with the sequence of pentagonal numbers starting: 1, 5, 12, 22, 35, ...^[37]

The factorial of five ${\displaystyle 5!=120}$ is multiply perfect like 28 and 496.^[42] 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).^[43] On its own, 31 is the first prime centered pentagonal number,^[44] and the fifth centered triangular number.^[45] 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).^[46] 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.^[47] 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, ...^[48] The first five members in this sequence add to 126, which is also the sixth pentagonal pyramidal number^[49] as well as the fifth ${\displaystyle {\mathcal {S}}}$-perfect Granville number.^[50] This is the third Granville number not to be perfect, and the only known such number with three distinct prime factors.^[51]

55 is the fifteenth discrete biprime,^[52] 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).^[53][54] Fifty-five is also the tenth Fibonacci number,^[55] whose digit sum is also 10, in its decimal representation. It is the tenth triangular number and the fourth that is doubly triangular,^[56] the fifth heptagonal number^[57] and fourth centered nonagonal number,^[58] and as listed above, the fifth square pyramidal number.^[39] The sequence of triangular ${\displaystyle n}$ that are powers of 10 is: 55, 5050, 500500, ...^[59] 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,^[60] 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 ${\displaystyle R(5,5)}$,^[61][62] and is a Schröder–Hipparchus number; the next and fifth such number is 197, the forty-fifth prime number^[24] that represents the number of ways of dissecting a heptagon into smaller polygons by inserting diagonals.^[63] A five-sided convex pentagon, on the other hand, has eleven ways of being subdivided in such manner.^{[lower-alpha 1]}

Magic figures

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.^[64] 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,^[65] 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,^[52] with an aliquot sum of 19 as well^[43] and equivalent to 1⁵ + 2⁴ + 3³ + 4² + 5¹.^[66] It is also the magic constant of the ${\displaystyle n-}$Queens Problem for ${\displaystyle n=5}$,^[67] the fifth octagonal number,^[68] 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.^[69] 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.^[70] In between these three Markov numbers is the tenth prime number 29^[24] 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).^[71] A magic constant of 505 is generated by a ${\displaystyle 10\times 10}$ normal magic square,^[70] where 10 is the fifth composite.^[72]

5 is also the value of the central cell the only non-trivial normal magic hexagon made of nineteen cells.^{[73][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.^[24] 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).^[43] The smallest value that the magic constant of a five-pointed magic pentagram can have using distinct integers is 24.^{[80][lower-alpha 4]}

Collatz conjecture

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 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}, where 16 is the smallest number with exactly five divisors,^[83] and one of only two numbers to have an aliquot sum of 15, the other being 33.^[43] Otherwise, 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, 53, 5, 2, 1} that are prime, including 1.^[84] Overall, thirteen numbers in the Collatz map for 15 back to 1 are composite,^[81] where the largest prime in the trajectory of 120 back to {4 ➙ 2 ➙ 1 ➙ 4 ➙ ...} is the sixteenth prime number, 53.^[24]

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.^[85] It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).^[86]

Generalizations

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.^[87] Meanwhile:

As a consequence of Fermat's little theorem and Euler's criterion, all squares are congruent to 0, 1, 4 (or −1) modulo 5.^[90] In particular, all integers ${\displaystyle n\geq 34}$ can be expressed as the sum of five non-zero squares.^[91][92]

Regarding Waring's problem, ${\displaystyle g(5)=37}$, where every natural number ${\displaystyle n\in \mathbb {N} }$ is the sum of at most thirty-seven fifth powers.^[93][94]

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 ${\displaystyle B}$.^{[95]: pp.1, 2} Aside from ${\displaystyle \{1\}}$, the five classes of Ramsey permutations are the classes of:^[95]: 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.^[95]: p.3

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 ${\displaystyle n}$ ⩾ ${\displaystyle 5}$.

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.^[96] Whereas the real numbers contain all five properties, the octonions are only alternative and power-associative. In comparison, 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.^[97] Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16.

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 K_3,3.^[100] 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.^[101][102] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.^[103] 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.^[104][105] 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.^[106] The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.^[107]

Polyhedra

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.^[108] 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,^[109] 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.^[116] 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.

Fourth dimension

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.^{[117]: 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.^[121] 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:^[122]

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

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.

Fifth dimension

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.^[123] 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).^[124]

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.^[125]

Finite simple groups

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.^[126] ${\displaystyle {\mathfrak {e}}_{8}}$ 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.^[127] 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.^[128][129] The smallest simple isomorphism found inside finite simple Lie groups is ${\displaystyle \mathrm {A_{5}} \cong A_{1}(4)\cong A_{1}(5)}$,^[130] where here ${\displaystyle \mathrm {A_{n}} }$ represents alternating groups and ${\displaystyle A_{n}(q)}$ classical Chevalley groups. In particular, the smallest non-solvable group is the alternating group on five letters, which is also the smallest simple non-abelian group.

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}.^{[131]: 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.^[132] 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.^[133] ${\displaystyle \mathrm {M} _{22}}$ 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.^{[131]: 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.^{[131]: 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.^{[131]: 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.^{[131]: 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.^[134] 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.^[135][136] On its own, ${\displaystyle \mathrm {HN} }$ can be represented using standard generators ${\displaystyle (a,b,ab)}$ that further dictate a condition where ${\displaystyle o([a,b])=5}$.^[137][138] This condition is also held by other generators that belong to the Tits group ${\displaystyle \mathrm {T} }$,^[139] the only finite simple group that is a non-strict group of Lie type that can also classify as sporadic (fifth-largest of all twenty-seven by order, too). 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 ${\displaystyle V_{2}}$^♮,^[140] which holds the friendly giant as its automorphism group.

In decimal

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 ${\displaystyle n}$.