5

This article is about the number. For the year, see AD 5. For other uses, see 5 (disambiguation), Number Five (disambiguation), and The Five (disambiguation).

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. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.

← 4 5 6 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers ← 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
Devanāgarī	५
Hebrew	ה
Khmer	៥
Telugu	౫
Malayalam	൫
Tamil	௫
Thai	๕

Evolution of the Arabic digit

 

The evolution of the modern Western digit for the numeral 5 cannot be traced back to the Indian system, as for the digits 1 to 4. The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. The Nagari and Punjabi took these digits and all came up with forms that were similar to a lowercase "h" rotated 180°. The Ghubar Arabs transformed the digit in several ways, producing from that were more similar to the digits 4 or 3 than to 5.^[1] It was from those digits that Europeans finally came up with the modern 5.

 

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

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

Mathematics

 

The first Pythagorean triple, with a hypotenuse of ${\displaystyle 5}$ 

Five is the third smallest prime number, and the second super-prime.^[2] It is the first safe prime,^[3] the first good prime,^[4] the first balanced prime,^[5] and the first of three known Wilson primes.^[6] Five is the second Fermat prime,^[2] the second Proth prime,^[7] and the third Mersenne prime exponent,^[8] as well as the third Catalan number^[9] and the third Sophie Germain prime.^[2] Notably, 5 is equal to the sum of the only consecutive primes 2 + 3 and it is the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7).^[10][11] It also forms the first pair of sexy primes with 11,^[12] which is the fifth prime number and Heegner number,^[13] as well as the first repunit prime in decimal; a base in-which five is also the first non-trivial 1-automorphic number.^[14] Five is the third factorial prime,^[15] and an alternating factorial.^[16] It is also an Eisenstein prime (like 11) with no imaginary part and real part of the form ${\displaystyle 3p-1}$ .^[2] In particular, five is the first congruent number, since it is the length of the hypotenuse of the smallest integer-sided right triangle.^[17]

Number theory

5 is the fifth Fibonacci number, being 2 plus 3.^[2] It is the only Fibonacci number that is equal to its position aside from 1, which is both the first and second Fibonacci numbers. 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). Whereas 5 is unique in the Fibonacci sequence, 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.

Figurate numbers and magic figures

In figurate numbers, 5 is a pentagonal number, with the sequence of pentagonal numbers starting: 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]

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 the fifth non-trivial 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]

 

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 ${\displaystyle 3}$  x ${\displaystyle 3}$  array has a magic constant ${\displaystyle M}$  of ${\displaystyle 15}$ , where the sums of its rows, columns, and diagonals are all equal to fifteen.^[51] 5 is also the value of the central cell the only non-trivial normal magic hexagon made of nineteen cells.^[52]

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).^[53][54] 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.^[55] It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).^[56]

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.^[57] Meanwhile:

• Every odd number greater than ${\displaystyle 1}$  is the sum of at most five prime numbers,^[58] and
• Every odd number greater than ${\displaystyle 5}$  is conjectured to be expressible as the sum of three prime numbers.^[59] Helfgott has provided a proof of this, also known as the odd Goldbach conjecture, that is already widely acknowledged by mathematicians as it still undergoes peer-review.

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 is a monochromatic subobject isomorphic to ${\displaystyle B}$ .^{[60]: pp.1, 2} Aside from ${\displaystyle \{1\}}$ , the five classes of Ramsey permutations are the class of identity permutations, the class of reversals, the class of increasing sequences of decreasing sequences, the class of decreasing sequences of increasing sequences, and the class of all permutations.^[60]: 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.^[60]: p.3

Inside the 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 properties.^[61] 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.^[62] Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16.

Geometry

 

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.^[63] 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.^[64]

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.^[65] 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.^[66][67] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.^[68] 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.^[69][70] 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.^[71] The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.^[72]

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: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.^[73] The dodecahedron in particular contains pentagonal faces, while the icosahedron, its dual polyhedron, has a vertex figure that is a regular pentagon. There are also five:

• Regular polyhedron compounds: the stella octangula, compound of five tetrahedra, compound of five cubes, compound of five octahedra, and compound of ten tetrahedra.^[74] Icosahedral symmetry ${\displaystyle \mathrm {I} _{h}}$  is isomorphic to the alternating group on five letters ${\displaystyle \mathrm {A} _{5}}$  of order 120, realized by actions on these uniform polyhedron compounds.

• Space-filling convex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium.^[75] 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.^[76] 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.^[77]

• Regular abstract polyhedra, which include the excavated dodecahedron and the dodecadodecahedron.^[78] 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.

There are also five semiregular prisms that are facets inside non-prismatic uniform four-dimensional figures: the triangular, pentagonal, hexagonal, octagonal, and decagonal prisms. Five uniform prisms and antiprisms contain pentagons or pentagrams: the pentagonal prism and antiprism, and the pentagrammic prism, antiprism, and crossed-antirprism.^[79]

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.^[80]

• 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 ${\displaystyle \{5\}\otimes \{5/3\}}$  and made of fifty tetrahedra, ten pentagrammic crossed antiprisms, ten pentagonal antiprisms, and fifty vertices.^[81]

• The grand antiprism, which is the only known non-Wythoffian construction of a uniform polychoron, is made of twenty pentagonal antiprisms and three hundred tetrahedra, with a total of one hundred vertices and five hundred edges.^[82]

• The abstract four-dimensional 57-cell is made of fifty-seven hemi-icosahedral cells, in-which five surround each edge.^[83] 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.^[84] 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: ${\displaystyle \mathrm {A} _{4}}$ , ${\displaystyle \mathrm {B} _{4}}$ , ${\displaystyle \mathrm {D} _{4}}$ , ${\displaystyle \mathrm {F} _{4}}$ , and ${\displaystyle \mathrm {H} _{4}}$ , accompanied by a fifth or sixth general group of unique 4-prisms of Platonic and Archimedean solids. All of these uniform 4-polytopes are generated from twenty-five uniform polyhedra, which include the five Platonic solids, fifteen Archimedean solids counting two enantiomorphic forms, and five prisms. 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.^[85] 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:^[86]

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

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 semiregular polytope, and has the rectified 5-cell as its vertex figure, which is one of only three semiregular 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.^[87] 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).^[88]

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.^[89]

Finite simple groups

There are five 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, ${\displaystyle {\mathfrak {g}}_{2}}$ , 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.^[90] ${\displaystyle {\mathfrak {e}}_{8}}$  is the largest of all five exceptional groups, with the other four as subgroups, and 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.^[91] 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 semiregular honeycomb, which includes the three-dimensional alternated cubic honeycomb.^[92][93] 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 ${\displaystyle \mathrm {A_{n}} }$  represents alternating groups and ${\displaystyle A_{n}(q)}$  classical Chevalley groups. The smallest alternating group that is simple is the alternating group on five letters.

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}.^[94]: 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.^[95] 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.^[96] ${\displaystyle \mathrm {M} _{22}}$  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.^[94]: 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.^{[94]: 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.^[94]: 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 the subject of the second generation of seven sporadic groups that are subquotients of the automorphism of the Leech lattice, Conway group ${\displaystyle \mathrm {Co} _{0}}$ .^{[94]: 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.^[97] 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.^[98][99] On its own, ${\displaystyle \mathrm {HN} }$  can be represented using standard generators ${\displaystyle (a,b,ab)}$  that further dictate a condition where ${\displaystyle o([a,b])=5}$ .^[100][101] This condition is also held by other generators that belong to the Tits group ${\displaystyle \mathrm {T} }$ ,^[102] 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 ${\displaystyle V_{2}}$ ^♮,^[103] which holds the friendly giant as its automorphism group.

Euler's identity

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 ${\displaystyle 0}$ .^{[104][105][106]}

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
5^x	5	25	125	625	3125	15625	78125	390625	1953125	9765625		48828125	244140625	1220703125	6103515625	30517578125
x^5	1	32	243	1024		7776	16807	32768	59049	100000		161051	248832	371293	537824	759375

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 ${\displaystyle n}$ .

Science

• The atomic number of boron.^[107]
• The number of appendages on most starfish, which exhibit pentamerism.^[108]
• The most destructive known hurricanes rate as Category 5 on the Saffir–Simpson hurricane wind scale.^[109]
• The most destructive known tornadoes rate an F-5 on the Fujita scale or EF-5 on the Enhanced Fujita scale.^[110]

Astronomy

• 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.^[111]
• The New General Catalogue object NGC 5, a magnitude 13 spiral galaxy in the constellation Andromeda.^[112]
• Messier object M5, a magnitude 7.0 globular cluster in the constellation Serpens.^[113]

Biology

• There are usually considered to be five senses (in general terms).
• The five basic tastes are sweet, salty, sour, bitter, and umami.^[114]
• Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.^[115]

Computing

• 5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.^[116]

Religion and culture

Hinduism

• The god Shiva has five faces^[117] 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.^{[clarification needed]}
• 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.

Christianity

• 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.^[118]

Gnosticism

• 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

Islam

• The Five Pillars of Islam^[119]
• Muslims pray to Allah five times a day^[120]
• 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.^[121]

Judaism

• 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").^[122]
• The book of Psalms is arranged into five books, paralleling the Five Books of Moses.^[123]
• 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.^[124]

Sikhism

• 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).^[125] Also, there are five deadly evils: kam (lust), krodh (anger), moh (attachment), lobh (greed), and ankhar (ego).

Daoism

• 5 Elements^[126]
• 5 Emperors^[127]

Other religions and cultures

• 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 religious significance in various faiths 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).^[128] 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.^[129] 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.^[130]

Art, entertainment, and media

Fictional entities

• James the Red Engine, a fictional character numbered 5.^[131]
• Johnny 5 is the lead character in the film Short Circuit (1986)^[132]
• Number Five is a character in Lorien Legacies^[133]
• 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^[134]
• 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 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^[135]
• 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)^[136]
• 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^[137]

Films

• 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^[138]
• The Fifth Element (1997), a science fiction film^[139]
• Fast Five (2011), the fifth installment of the Fast and Furious film series.^[140]
• 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^[141]

Music

• Modern musical notation uses a musical staff made of five horizontal lines.^[142]
• A scale with five notes per octave is called a pentatonic scale.^[143]
• A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.^[144]
• 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.^[145]
• Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.

Groups

• Five (group), a UK Boy band^[146]
• The Five (composers), 19th-century Russian composers^[147]
• 5 Seconds of Summer, pop band that originated in Sydney, Australia
• Five Americans, American rock band active 1965–1969^[148]
• 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^[149]
• Maroon 5, American pop rock band that originated in Los Angeles, California^[150]
• MC5, American punk rock band^[151]
• Pentatonix, a Grammy-winning a cappella group originated in Arlington, Texas^[152]
• The 5th Dimension, American pop vocal group, active 1977–present^[153]
• 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^[154]
• 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^[155]
• 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^[156]
• Fifth Harmony, an American girl group.^[157]
• Ben Folds Five, an American alternative rock trio, 1993–2000, 2008 and 2011–2013^[158]
• R5 (band), an American pop and alternative rock group, 2009–2018^[159]

Other

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

Television

Stations

• Channel 5 (UK), a television channel that broadcasts in the United Kingdom^[160]
• 5 (TV channel) (formerly known as ABC 5 and TV5) (DWET-TV channel 5 In Metro Manila) a television network in the Philippines.^[161]

Series

• Babylon 5, a science fiction television series^[162]
• 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^[163]
• Hi-5 (UK 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^[164]
• 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.^[165]

Literature

• 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^[166]

Sports

• 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).^[167]
• In AFL Women's, the top level of women's Australian rules football, each team is allowed 5 "interchanges" (substitute players), who can be freely substituted at any time.
• In baseball scorekeeping, the number 5 represents the third baseman's position.
• In basketball:
  • The number 5 is used to represent the position of center.
  • Each team has five players on the court at a given time. Thus, the phrase "five on five" is commonly used to describe standard competitive basketball.^[168]
  • The "5-second rule" refers to several related rules designed to promote continuous play. In all cases, violation of the rule results in a turnover.
  • Under the FIBA (used for all international play, and most non-US leagues) and NCAA women's rule sets, a team begins shooting bonus free throws once its opponent has committed five personal fouls in a quarter.
  • Under the FIBA rules, A player fouls out and must leave the game after committing five fouls
• Five-a-side football is a variation of association football in which each team fields five players.^[169]
• In ice hockey:
  • A major penalty lasts five minutes.^[170]
  • There are five different ways that a player can score a goal (teams at even strength, team on the power play, team playing shorthanded, penalty shot, and empty net).^[171]
  • The area between the goaltender's legs is known as the five-hole.^[172]
• In most rugby league competitions, the starting left wing wears this number. An exception is the Super League, which uses static squad numbering.
• In rugby union:
  • A try is worth 5 points.^[173]
  • One of the two starting lock forwards wears number 5, and usually jumps at number 4 in the line-out.
  • In the French variation of the bonus points system, a bonus point in the league standings is awarded to a team that loses by 5 or fewer points.

Technology

 

5 as a resin identification code, used in recycling.

• 5 is the most common number of gears for automobiles with manual transmission.^[174]
• In radio communication, the term "Five by five" is used to indicate perfect signal strength and clarity.^[175]
• On almost all devices with a numeric keypad such as telephones, computers, etc., the 5 key has a raised dot or raised bar to make dialing easier. Persons who are blind or have low vision find it useful to be able to feel the keys of a telephone. All other numbers can be found with their relative position around the 5 button (on computer keyboards, the 5 key of the numpad has the raised dot or bar, but the 5 key that shifts with % does not).^[176]
• On most telephones, the 5 key is associated with the letters J, K, and L,^[177] but on some of the BlackBerry phones, it is the key for G and H.
• The Pentium, coined by Intel Corporation, is a fifth-generation x86 architecture microprocessor.^[178]
• The resin identification code used in recycling to identify polypropylene.^[179]

Miscellaneous fields

 

International maritime signal flag for 5

 

St. Petersburg Metro, Line 5

 

The fives of all four suits in playing cards

Five can refer to:

• "Give me five" is a common phrase used preceding a high five.
• An informal term for the British Security Service, MI5.
• Five babies born at one time are quintuplets. The most famous set of quintuplets were the Dionne quintuplets born in the 1930s.^[180]
• In the United States legal system, the Fifth Amendment to the United States Constitution can be referred to in court as "pleading the fifth", absolving the defendant from self-incrimination.^[181]
• Pentameter is verse with five repeating feet per line; iambic pentameter was the most popular form in Shakespeare.^[182]
• Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth)^[183]
• The designation of an Interstate Highway (Interstate 5) that runs from San Diego, California to Blaine, Washington.^[184] In addition, all major north-south Interstate Highways in the United States end in 5.^[185]
• In the computer game Riven, 5 is considered a holy number, and is a recurring theme throughout the game, appearing in hundreds of places, from the number of islands in the game to the number of bolts on pieces of machinery.
• The Garden of Cyrus (1658) by Sir Thomas Browne is a Pythagorean discourse based upon the number 5.
• The holy number of Discordianism, as dictated by the Law of Fives.^[186]
• The number of Justices on the Supreme Court of the United States necessary to render a majority decision.^[187]
• The number of dots in a quincunx.^[188]
• The number of permanent members with veto power on the United Nations Security Council.^[189]
• The number of Korotkoff sounds when measuring blood pressure^[190]
• The drink Five Alive is named for its five ingredients. The drink punch derives its name after the Sanskrit पञ्च (pañc) for having five ingredients.^[191]
• The Keating Five were five United States Senators accused of corruption in 1989.^[192]
• The Inferior Five: Merryman, Awkwardman, The Blimp, White Feather, and Dumb Bunny. DC Comics parody superhero team.^[193]
• No. 5 is the name of the iconic fragrance created by Coco Chanel.^[194]
• The Committee of Five was delegated to draft the United States Declaration of Independence.^[195]
• The five-second rule is a commonly used rule of thumb for dropped food.^[196]
• 555 95472, usually referred to simply as 5, is a minor male character in the comic strip Peanuts.^[197]

See also

•  Mathematics portal

• List of highways numbered 5

References

1. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.65
2. Weisstein, Eric W. "5". mathworld.wolfram.com. Retrieved 2020-07-30.
3. 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.
4. Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. 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.
6. Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
7. Sloane, N. J. A. (ed.). "Sequence A080076 (Proth primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
8. Weisstein, Eric W. "Mersenne Prime". mathworld.wolfram.com. Retrieved 2020-07-30.
9. Weisstein, Eric W. "Catalan Number". mathworld.wolfram.com. Retrieved 2020-07-30.
10. Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
11. Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
12. 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.
13. Sloane, N. J. A. (ed.). "Sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-20.
14. Sloane, N. J. A. (ed.). "Sequence A003226 (Automorphic numbers: m^2 ends with m.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26.
15. 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.
16. Weisstein, Eric W. "Twin Primes". mathworld.wolfram.com. Retrieved 2020-07-30.
17. Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
18. Weisstein, Eric W. "Perrin Sequence". mathworld.wolfram.com. Retrieved 2020-07-30.
19. Weisstein, Eric W. "Sierpiński Number of the First Kind". mathworld.wolfram.com. Retrieved 2020-07-30.
20. Sloane, N. J. A. (ed.). "Sequence A019434 (Fermat primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21.
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.
22. 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.
23. 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.
24. 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.
25. 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.
26. Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 84–86. ISBN 0-387-20860-7.
27. Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
28. Sloane, N. J. A. (ed.). "Sequence A076046 (Ramanujan-Nagell numbers: the triangular numbers...which are also of the form 2^b - 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
29. Sloane, N. J. A. (ed.). "Sequence A000225 (... (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-13.
30. 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.
31. 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.
32. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
33. 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.
34. 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.
35. 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.
36. Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
37. Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
38. Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
39. 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.
40. 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.
41. 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.
42. Sloane, N. J. A. (ed.). "Sequence A001065". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-11.
43. Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
44. Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
45. Conrad, Keith E. "Example of Mordell's Equation" (PDF) (Professor Notes). University of Connecticut (Homepage). p. 10. S2CID 5216897.
46. 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.
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180. Corporation, Bonnier (1937). Popular Science. Bonnier Corporation. p. 32. ...another picture of one of the world's most famous babies was made. Fred Davis is official photographer of the Dionne quintuplets...

August 26, 2023