5

This article is about the number. For the years, see 5 BC and 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] 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][25][2] This is the prime index of the third Mersenne prime and second double Mersenne prime 127,^[26] as well as the third double Mersenne prime exponent for the number 2,147,483,647,^[26] 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.^[27][28] 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.^[29][30] 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.^[31][32][33] 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.^[34][35] The fifth Mersenne prime, 8191,^[25] splits into 4095 and 4096, with the latter being the fifth superperfect number^[36] and the sixth power of four, 4⁶.

Figurate numbers

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

• 5 is a centered tetrahedral number: 1, 5, 15, 35, 69, ...^[38] 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, ...^[39] 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, ...^[40] 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.^[41]

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.^[a]

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^[b] 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][c] 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][d]

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 ^[e] 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

Unsolved problem in mathematics:

Is 5 the only odd untouchable number?

(more unsolved problems in mathematics)

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:

• Every odd number greater than ${\displaystyle 1}$  is the sum of at most five prime numbers,^[88] and
• Every odd number greater than ${\displaystyle 5}$  is conjectured to be expressible as the sum of three prime numbers. Helfgott has provided a proof of this^[89] (also known as the odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes peer-review.

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

• Identity permutations, and reversals
• Increasing sequences of decreasing sequences, and decreasing sequences of increasing sequences
• All permutations

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.

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.^[98] 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.^[99]

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:

• Regular polyhedron compounds: the compound of five tetrahedra, compound of ten tetrahedra, compound of five cubes, compound of five octahedra, and stella octangula.^[109] 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 (aside from the regular compound of two tetrahedra). All fifteen mirror planes of ${\displaystyle \mathrm {I} _{h}}$  pass through the edges of a regular spherical compound of five octahedra, whose sets of three orthogonal great circles use five colors.^{[f][110][111]}
• Space-filling convex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium.^[112] 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.^[113] 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.^[114]
• Regular abstract polyhedra, which include the excavated dodecahedron and the dodecadodecahedron.^[115] They have combinatorial symmetries transitive on flags of their elements, with topologies equivalent to that of toroids and the ability to tile the hyperbolic plane.

Moreover, 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}

• 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.^{[117]: p.124}
• 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.^[118]
• The abstract four-dimensional 57-cell is made of fifty-seven hemi-icosahedral cells, in-which five surround each edge.^[119] 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.^[120] The skeleton of the hemi-dodecahedron is the Petersen graph.

Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora based on only twenty-five uniform polyhedra: ${\displaystyle \mathrm {A} _{4}}$ , ${\displaystyle \mathrm {B} _{4}}$ , ${\displaystyle \mathrm {D} _{4}}$ , ${\displaystyle \mathrm {F} _{4}}$ , 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.

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 ${\displaystyle 0}$ .^{[141][142][143]}

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

Astronomy

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

Biology

• There are usually considered to be five senses (in general terms); the five basic tastes are sweet, salty, sour, bitter, and umami.^[144]
• Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.^[145]
• Five is the number of appendages on most starfish, which exhibit pentamerism.^[146]

Computing

• 5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.^[147]

Literature

Poetry

A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.^[148]

Music

• Modern musical notation uses a musical staff made of five horizontal lines.^[149]
• A scale with five notes per octave is called a pentatonic scale.^[150]
• A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.^[151]
• In harmonics, the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A major triad chord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third.
• Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.

Religion and mysticism

Judaism

• The Book of Numbers is one of five books in the Torah; the others being the books of Genesis, Exodus, Leviticus, and Deuteronomy.

They are collectively called the Five Books of Moses, the Pentateuch (Greek for "five containers", referring to the scroll cases in which the books were kept), or Humash (חומש, Hebrew for "fifth").^[152]

• 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.^[153]

Christianity

• There are traditionally five wounds of Jesus Christ in Christianity: the nail wounds in Christ's two hands, the nail wounds in Christ's two feet, and the Spear Wound of Christ (respectively at the four extremities of the body, and the head).^[154]

Islam

• The Five Pillars of Islam.^[155]

Gnosticism

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

Elements

• According to ancient Greek philosophers such as Aristotle, the universe is made up of five classical elements: water, earth, air, fire, and ether. This concept was later adopted by medieval alchemists and more recently by practitioners of Neo-Pagan religions such as Wicca.
• There are five elements in the universe according to Hindu cosmology: dharti, agni, jal, vayu evam akash (earth, fire, water, air and space, respectively).
• The 5 Elements of traditional Chinese Wuxing.^[156]
• In East Asian tradition, there are five elements: (water, fire, earth, wood, and metal).^[157] 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.^[158] 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.

Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth), as a union of these.^[159] The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca. In numerology, 5 or a series of 555, is often associated with change, evolution, love and abundance.^{[citation needed]}

Miscellaneous

 

The fives of all four suits in playing cards

• "Give me five" is a common phrase used preceding a high five.
• The Olympic Games have five interlocked rings as their symbol, representing the number of inhabited continents represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).^[160]
• The number of dots in a quincunx.^[161]

See also

•  Mathematics portal

5 (disambiguation)

Notes

1.  
2. ${\displaystyle {\begin{bmatrix}17&24&1&8&15\\23&5&7&14&16\\4&6&13&20&22\\10&12&19&21&3\\11&18&25&2&9\end{bmatrix}}}$ 
3.  
4.  
5.  
6.  

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 A007540 (Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
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 A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-08.
25. 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.
26. 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.
27. Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 84–86. ISBN 0-387-20860-7.
28. Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
29. 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.
30. 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.
31. 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.
32. 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.
33. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
34. 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.
35. 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.
36. 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.
37. Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
38. Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
39. Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
40. 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.
41. 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.
42. 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.
43. Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-11.
44. Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
45. Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
46. Conrad, Keith E. "Example of Mordell's Equation" (PDF) (Professor Notes). University of Connecticut (Homepage). p. 10. S2CID 5216897.
47. 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.
48. Sloane, N. J. A. (ed.). "Sequence A000332 (Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-14.
49. Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-28.
50. Sloane, N. J. A. (ed.). "Sequence A118372 (S-perfect numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-28.
51. de Koninck, Jean-Marie (2008). Those Fascinating Numbers. Translated by de Koninck, J. M. Providence, RI: American Mathematical Society. p. 40. ISBN 978-0-8218-4807-4. MR 2532459. OCLC 317778112.
52. Sloane, N. J. A. (ed.). "Sequence A006881 (Squarefree semiprimes: Numbers that are the product of two distinct primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
53. Sloane, N. J. A. (ed.). "Sequence A216071 (Brocard's problem: positive integers m such that m^2 equal to n! + 1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-09.
54. Sloane, N. J. A. (ed.). "Sequence A085692 (Brocard's problem: squares which can be written as n!+1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-09.
55. Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci numbers: F(n) is F(n-1) + F(n-2) with F(0) equal to 0 and F(1) equal to 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
56. Sloane, N. J. A. (ed.). "Sequence A002817 (Doubly triangular numbers: a(n) equal to n*(n+1)*(n^2+n+2)/8.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
57. Sloane, N. J. A. (ed.). "Sequence A000566 (Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
58. Sloane, N. J. A. (ed.). "Sequence A060544 (Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1) equal to 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
59. Sloane, N. J. A. (ed.). "Sequence A037156 (a(n) equal to 10^n*(10^n+1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
    a(0) = 1 = 1 * 1 = 1
    a(1) = 1 + 2 + ...... + 10 = 11 * 5 = 55
    a(2) = 1 + 2 + .... + 100 = 101 * 50 = 5050
    a(3) = 1 + 2 + .. + 1000 = 1001 * 500 = 500500
    ...
60. Sloane, N. J. A. (ed.). "Sequence A006886 (Kaprekar numbers...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
61. Sloane, N. J. A. (ed.). "Sequence A120414 (Conjectured Ramsey number R(n,n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
62. Sloane, N. J. A. (ed.). "Sequence A212954 (Triangle read by rows: two color Ramsey numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
63. Sloane, N. J. A. (ed.). "Sequence A001003 (Schroeder's second problem; ... also called super-Catalan numbers or little Schroeder numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
64. William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 2022-07-14.
65. Sloane, N. J. A. (ed.). "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
66. Sloane, N. J. A. (ed.). "Sequence A003101 (a(n) as Sum_{k equal to 1..n} (n - k + 1)^k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
67. Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
68. Sloane, N. J. A. (ed.). "Sequence A000567 (Octagonal numbers: n*(3*n-2). Also called star numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
69. "Sloane's A008277 :Triangle of Stirling numbers of the second kind". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-12-24.
70. Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
71. Sloane, N. J. A. (ed.). "Sequence A000162 (Number of 3-dimensional polyominoes (or polycubes) with n cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
72. Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-25.
73. Trigg, C. W. (February 1964). "A Unique Magic Hexagon". Recreational Mathematics Magazine. Retrieved 2022-07-14.
74. Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-10.
75. Sloane, N. J. A. (ed.). "Sequence A000928 (Irregular primes: primes p such that at least one of the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) is divisible by p.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
76. Sloane, N. J. A. (ed.). "Sequence A189683 (Irregular pairs (p,2k) ordered by increasing k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
77. Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-10.
78. Sloane, N. J. A. (ed.). "Sequence A000203 (a(n) is sigma(n), the sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-14.
79. Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-14.
80. Gardner, Martin (1989). Mathematical Carnival. Mathematical Games (5th ed.). Washington, D.C.: Mathematical Association of America. pp. 56–58. ISBN 978-0-88385-448-8. OCLC 20003033. Zbl 0684.00001.
81. Sloane, N. J. A. (ed.). "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-01-24.
82. Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-24.

    "Table of n, a(n) for n = 1..10000"

83. Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-06.
84. Sloane, N. J. A. (ed.). "Sequence A286380 (a(n) is the minimum number of iterations of the reduced Collatz function R required to yield 1. The function R (A139391) is defined as R(k) equal to (3k+1)/2^r, with r as large as possible.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-18.
85. Sloane, N. J. A. (ed.). "Sequence A003079 (One of the basic cycles in the x->3x-1 (x odd) or x/2 (x even) problem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-24.

    {5 ➙ 14 ➙ 7 ➙ 20 ➙ 10 ➙ 5 ➙ ...}.

86. Sloane, N. J. A. (ed.). "3x-1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-01-24.
87. Pomerance, Carl (2012). "On Untouchable Numbers and Related Problems" (PDF). Dartmouth College: 1. S2CID 30344483.
88. Tao, Terence (March 2014). "Every odd number greater than 1 is the sum of at most five primes" (PDF). Mathematics of Computation. 83 (286): 997–1038. doi:10.1090/S0025-5718-2013-02733-0. MR 3143702. S2CID 2618958.
89. Helfgott, Harald Andres (2014). "The ternary Goldbach problem" (PDF). In Jang, Sun Young (ed.). Seoul International Congress of Mathematicians Proceedings. Vol. 2. Seoul, KOR: Kyung Moon SA. pp. 391–418. ISBN 978-89-6105-805-6. OCLC 913564239.
90. Sellers, James A. (2013). "An unexpected congruence modulo 5 for 4-colored generalized Frobenius partitions". J. Indian Math. Soc. Pune, IMD: Indian Mathematical Society. New Series (Special Issue): 99. arXiv:1302.5708. Bibcode:2013arXiv1302.5708S. MR 0157339. S2CID 116931082. Zbl 1290.05015.
91. Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1980). An Introduction to the Theory of Numbers (5th ed.). New York, NY: John Wiley. pp. 144, 145. ISBN 978-0-19-853171-5.
92. Sloane, N. J. A. (ed.). "Sequence A047701 (All positive numbers that are not the sum of 5 nonzero squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-20.

    Only twelve integers up to 33 cannot be expressed as the sum of five non-zero squares: {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33} where 2, 3 and 7 are the only such primes without an expression.

93. Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem: g(n) equal to 2^n + floor((3/2)^n) - 2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-20.
94. Hilbert, David (1909). "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)". Mathematische Annalen (in German). 67 (3): 281–300. doi:10.1007/bf01450405. JFM 40.0236.03. MR 1511530. S2CID 179177986.
95. Böttcher, Julia; Foniok, Jan (2013). "Ramsey Properties of Permutations". The Electronic Journal of Combinatorics. 20 (1): P2. arXiv:1103.5686v2. doi:10.37236/2978. S2CID 17184541. Zbl 1267.05284.
96. Kantor, I. L.; Solodownikow, A. S. (1989). Hypercomplex Numbers: An Elementary Introduction to Algebras. Translated by Shenitzer., A. New York, NY: Springer-Verlag. pp. 109–110. ISBN 978-1-4612-8191-7. OCLC 19515061. S2CID 60314285.
97. Imaeda, K.; Imaeda, M. (2000). "Sedenions: algebra and analysis". Applied Mathematics and Computation. Amsterdam, Netherlands: Elsevier. 115 (2): 77–88. doi:10.1016/S0096-3003(99)00140-X. MR 1786945. S2CID 32296814. Zbl 1032.17003.
98. Sarhangi, Reza (2012). "Interlocking Star Polygons in Persian Architecture: The Special Case of the Decagram in Mosaic Designs" (PDF). Nexus Network Journal. 14 (2): 350. doi:10.1007/s00004-012-0117-5. S2CID 124558613.
99. Coxeter, H. S. M.; du Val, P.; et al. (1982). The Fifty-Nine Icosahedra (1 ed.). New York: Springer-Verlag. pp. 7, 8. doi:10.1007/978-1-4613-8216-4. ISBN 978-0-387-90770-3. OCLC 8667571. S2CID 118322641.
100. Burnstein, Michael (1978). "Kuratowski-Pontrjagin theorem on planar graphs". Journal of Combinatorial Theory. Series B. 24 (2): 228–232. doi:10.1016/0095-8956(78)90024-2.
101. Holton, D. A.; Sheehan, J. (1993). The Petersen Graph. Cambridge University Press. pp. 9.2, 9.5 and 9.9. ISBN 0-521-43594-3.
102. Alon, Noga; Grytczuk, Jaroslaw; Hałuszczak, Mariusz; Riordan, Oliver (2002). "Nonrepetitive colorings of graphs" (PDF). Random Structures & Algorithms. 2 (3–4): 337. doi:10.1002/rsa.10057. MR 1945373. S2CID 5724512. A coloring of the set of edges of a graph G is called non-repetitive if the sequence of colors on any path in G is non-repetitive...In Fig. 1 we show a non-repetitive 5-coloring of the edges of P... Since, as can easily be checked, 4 colors do not suffice for this task, we have π(P) = 5.
103. Royle, G. "Cubic Symmetric Graphs (The Foster Census)." 2008-07-20 at the Wayback Machine
104. de Grey, Aubrey D.N.J. (2018). "The Chromatic Number of the Plane is At Least 5". Geombinatorics. 28: 5–18. arXiv:1804.02385. MR 3820926. S2CID 119273214.
105. Exoo, Geoffrey; Ismailescu, Dan (2020). "The Chromatic Number of the Plane is At Least 5: A New Proof". Discrete & Computational Geometry. New York, NY: Springer. 64: 216–226. arXiv:1805.00157. doi:10.1007/s00454-019-00058-1. MR 4110534. S2CID 119266055. Zbl 1445.05040.
106. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 227–236. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
107. Grünbaum, Branko; Shephard, Geoffrey C. (1987). "Tilings by polygons". Tilings and Patterns. New York: W. H. Freeman and Company. ISBN 978-0-7167-1193-3. MR 0857454. Section 9.3: "Other Monohedral tilings by convex polygons".
108. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61
109. Skilling, John (1976). "Uniform Compounds of Uniform Polyhedra". Mathematical Proceedings of the Cambridge Philosophical Society. 79 (3): 447–457. Bibcode:1976MPCPS..79..447S. doi:10.1017/S0305004100052440. MR 0397554. S2CID 123279687.
110. Hart, George W. (1998). "Icosahedral Constructions" (PDF). In Sarhangi, Reza (ed.). Bridges: Mathematical Connections in Art, Music, and Science. Winfield, Kansas: The Bridges Organization. p. 196. ISBN 978-0-9665201-0-1. OCLC 59580549. S2CID 202679388.
111. Hart, George W. "Symmetry Planes". Virtual Polyhedra (The Encyclopedia of Polyhedra). Retrieved 2023-09-27.

     "They can be colored as five sets of three mutually orthogonal planes" where the "fifteen planes divide the sphere into 120 Möbius triangles."

112. Kepler, Johannes (2010). The Six-Cornered Snowflake. Paul Dry Books. Footnote 18, p. 146. ISBN 978-1-58988-285-0.
113. Alexandrov, A. D. (2005). "8.1 Parallelohedra". Convex Polyhedra. Springer. pp. 349–359.
114. Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 2022-11-26. Retrieved 2023-01-12.
115. Wills, J. M. (1987). "The combinatorially regular polyhedra of index 2". Aequationes Mathematicae. 34 (2–3): 206–220. doi:10.1007/BF01830672. S2CID 121281276.
116. Har'El, Zvi (1993). "Uniform Solution for Uniform Polyhedra" (PDF). Geometriae Dedicata. Netherlands: Springer Publishing. 47: 57–110. doi:10.1007/BF01263494. MR 1230107. S2CID 120995279. Zbl 0784.51020.

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

117. H. S. M. Coxeter (1973). Regular Polytopes (3rd ed.). New York: Dover Publications, Inc. pp. 1–368. ISBN 978-0-486-61480-9.
118. John Horton Conway; Heidi Burgiel; Chaim Goodman-Strauss (2008). The Symmetries of Things. A K Peters/CRC Press. ISBN 978-1-56881-220-5. Chapter 26: "The Grand Antiprism"
119. Coxeter, H. S. M. (1982). "Ten toroids and fifty-seven hemidodecahedra". Geometriae Dedicata. 13 (1): 87–99. doi:10.1007/BF00149428. MR 0679218. S2CID 120672023..
120. Coxeter, H. S. M (1984). "A Symmetrical Arrangement of Eleven Hemi-Icosahedra". Annals of Discrete Mathematics. North-Holland Mathematics Studies. 87 (20): 103–114. doi:10.1016/S0304-0208(08)72814-7. ISBN 978-0-444-86571-7.
121. McMullen, Peter; Schulte, Egon (2002). Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications. Vol. 92. Cambridge: Cambridge University Press. pp. 162–164. doi:10.1017/CBO9780511546686. ISBN 0-521-81496-0. MR 1965665. S2CID 115688843.
122. Edge, William L. (1978). "Bring's curve". Journal of the London Mathematical Society. London: London Mathematical Society. 18 (3): 539–545. doi:10.1112/jlms/s2-18.3.539. ISSN 0024-6107. MR 0518240. S2CID 120740706. Zbl 0397.51013.
123. H.S.M. Coxeter (1956). "Regular Honeycombs in Hyperbolic Space". p. 168. CiteSeerX 10.1.1.361.251.
124. H. S. M. Coxeter (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. pp. 144–146. doi:10.2307/3617711. ISBN 978-0-521-39490-1. JSTOR 3617711. S2CID 116900933. Zbl 0732.51002.
125. Dixon, A. C. (March 1908). "The Conic through Five Given Points". The Mathematical Gazette. The Mathematical Association. 4 (70): 228–230. doi:10.2307/3605147. JSTOR 3605147. S2CID 125356690.
126. Baez, John C.; Huerta, John (2014). "G₂ and the rolling ball". Trans. Amer. Math. Soc. 366 (10): 5257–5293. doi:10.1090/s0002-9947-2014-05977-1. MR 3240924. S2CID 50818244.
127. Baez, John C. (2018). "From the Icosahedron to E₈". London Math. Soc. Newsletter. 476: 18–23. arXiv:1712.06436. MR 3792329. S2CID 119151549. Zbl 1476.51020.
128. H. S. M. Coxeter (1998). "Seven Cubes and Ten 24-Cells" (PDF). Discrete & Computational Geometry. 19 (2): 156–157. doi:10.1007/PL00009338. S2CID 206861928. Zbl 0898.52004.
129. Thorold Gosset (1900). "On the regular and semi-regular figures in space of n dimensions" (PDF). Messenger of Mathematics. 29: 43–48. JFM 30.0494.02.
130. Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford: Clarendon Press. p. xv. ISBN 978-0-19-853199-9. MR 0827219. OCLC 12106933. S2CID 117473588. Zbl 0568.20001.
131. Robert L. Griess, Jr. (1998). Twelve Sporadic Groups. Springer Monographs in Mathematics. Berlin: Springer-Verlag. pp. 1−169. doi:10.1007/978-3-662-03516-0. ISBN 978-3-540-62778-4. MR 1707296. S2CID 116914446. Zbl 0908.20007.
132. Jansen, Christoph (2005). "The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups". LMS Journal of Computation and Mathematics. London Mathematical Society. 8: 123–124. doi:10.1112/S1461157000000930. MR 2153793. S2CID 121362819. Zbl 1089.20006.
133. Cameron, Peter J. (1992). "Chapter 9: The geometry of the Mathieu groups" (PDF). Projective and Polar Spaces. University of London, Queen Mary and Westfield College. p. 139. ISBN 978-0-902-48012-4. S2CID 115302359.
134. Luis J. Boya (2011-01-16). "Introduction to Sporadic Groups". Symmetry, Integrability and Geometry: Methods and Applications. 7: 13. arXiv:1101.3055. Bibcode:2011SIGMA...7..009B. doi:10.3842/SIGMA.2011.009. S2CID 16584404.
135. Lux, Klaus; Noeske, Felix; Ryba, Alexander J. E. (2008). "The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2". Journal of Algebra. Amsterdam: Elsevier. 319 (1): 320–335. doi:10.1016/j.jalgebra.2007.03.046. MR 2378074. S2CID 120706746. Zbl 1135.20007.
136. Wilson, Robert A. (2009). "The odd local subgroups of the Monster". Journal of Australian Mathematical Society (Series A). Cambridge: Cambridge University Press. 44 (1): 12–13. doi:10.1017/S1446788700031323. MR 0914399. S2CID 123184319. Zbl 0636.20014.
137. Wilson, R.A (1998). "Chapter: An Atlas of Sporadic Group Representations" (PDF). The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249). Cambridge: Cambridge University Press. p. 267. doi:10.1017/CBO9780511565830.024. ISBN 978-0-511-56583-0. OCLC 726827806. S2CID 59394831. Zbl 0914.20016.
138. Nickerson, S.J.; Wilson, R.A. (2011). "Semi-Presentations for the Sporadic Simple Groups". Experimental Mathematics. Oxfordshire: Taylor & Francis. 14 (3): 367. doi:10.1080/10586458.2005.10128927. MR 2172713. S2CID 13100616. Zbl 1087.20025.
139. Wilson, R.A.; Parker, R.A.; Nickerson, S.J.; Bray, J.N. (1999). "Exceptional group ²F₄(2)', Tits group T". ATLAS of Finite Group Representations.
140. Ryba, A. J. E. (1996). "A natural invariant algebra for the Harada-Norton group". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge: Cambridge University Press. 119 (4): 597–614. Bibcode:1996MPCPS.119..597R. doi:10.1017/S0305004100074454. MR 1362942. S2CID 119931824. Zbl 0851.20034.
141. Wilson, Robin (2018). Euler's Pioneering Equation: The most beautiful theorem in mathematics. Oxford, UK: Oxford University Press. ISBN 978-0-192-51406-6. OCLC 990970269.
142. Paulos, John Allen (1992). Beyond Numeracy: An Uncommon Dictionary of Mathematics. New York, NY: Penguin Books. p. 117. ISBN 0-14-014574-5. OCLC 26361981.
143. Gallagher, James (13 February 2014). "Mathematics: Why the brain sees maths as beauty". BBC News Online. British Broadcasting Corporation (BBC). Retrieved 2023-06-02.
144. Marcus, Jacqueline B. (2013-04-15). Culinary Nutrition: The Science and Practice of Healthy Cooking. Academic Press. p. 55. ISBN 978-0-12-391883-3. There are five basic tastes: sweet, salty, sour, bitter and umami...
145. Kisia, S. M. (2010), Vertebrates: Structures and Functions, Biological Systems in Vertebrates, CRC Press, p. 106, ISBN 978-1-4398-4052-8, The typical limb of tetrapods is the pentadactyl limb (Gr. penta, five) that has five toes. Tetrapods evolved from an ancestor that had limbs with five toes. ... Even though the number of digits in different vertebrates may vary from five, vertebrates develop from an embryonic five-digit stage.
146. Cinalli, G.; Maixner, W. J.; Sainte-Rose, C. (2012-12-06). Pediatric Hydrocephalus. Springer Science & Business Media. p. 19. ISBN 978-88-470-2121-1. The five appendages of the starfish are thought to be homologous to five human buds
147. Pozrikidis, Constantine (2012-09-17). XML in Scientific Computing. CRC Press. p. 209. ISBN 978-1-4665-1228-3. 5 5 005 ENQ (enquiry)
148. Veith (Jr.), Gene Edward; Wilson, Douglas (2009). Omnibus IV: The Ancient World. Veritas Press. p. 52. ISBN 978-1-932168-86-0. The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter)
149. "STAVE | meaning in the Cambridge English Dictionary". dictionary.cambridge.org. Retrieved 2020-08-02. the five lines and four spaces between them on which musical notes are written
150. Ricker, Ramon (1999-11-27). Pentatonic Scales for Jazz Improvisation. Alfred Music. p. 2. ISBN 978-1-4574-9410-9. Pentatonic scales, as used in jazz, are five note scales
151. Danneley, John Feltham (1825). An Encyclopaedia, Or Dictionary of Music ...: With Upwards of Two Hundred Engraved Examples, the Whole Compiled from the Most Celebrated Foreign and English Authorities, Interspersed with Observations Critical and Explanatory. editor, and pub. are the perfect fourth, perfect fifth, and the octave
152. Pelaia, Ariela. "Judaism 101: What Are the Five Books of Moses?". Learn Religions. Retrieved 2020-08-03.

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