5

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.
In mathematics.
formula_1 is the third smallest prime number, and the second super-prime. It is the first safe prime, the first good prime, the first balanced prime, and the first of three known Wilson primes. Five is the second Fermat prime and the third Mersenne prime exponent, as well as the third Catalan number, and the third Sophie Germain prime. Notably, 5 is equal to the sum of the "only" consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7). It is also part of the first pair of sexy primes, with the fifth prime number and first prime repunit in decimal, 11. Five is the third factorial prime, an alternating factorial, and an Eisenstein prime with no imaginary part and real part of the form formula_2 − formula_3. In particular, five is the first congruent number, since it is the length of the hypotenuse of the smallest integer-sided right triangle.
Five is the second Fermat prime of the form formula_4+ formula_3, and more generally the second , formula_6+ formula_3. There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537. 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 31 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. Apropos, 31 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 formula_8-sided polygons, and equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem.
The number 5 is the fifth Fibonacci number, being 2 plus 3. 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), ... ( 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.
5 is the third Mersenne prime exponent of the form formula_9 − formula_3, which yields formula_11: the prime index of the third Mersenne prime and second double Mersenne prime 127, as well as the third double Mersenne prime exponent for the number 2,147,483,647, 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 formula_12 = 223058...93951 − 1 too large to compute with current computers. In a related sequence, the first 5 terms in the sequence of Catalan–Mersenne numbers formula_13 are the only known prime terms, with a sixth possible candidate in the order of 101037.7094. These prime sequences are conjectured to be prime up to a certain limit.
Every odd number greater than formula_3 is the sum of at most five prime numbers, and every odd number greater than formula_1 is conjectured to be expressible as the sum of three prime numbers. Helfgott has provided a proof of the latter, also known as the odd Goldbach conjecture, that is already widely acknowledged by mathematicians as it still undergoes peer-review.
There are a total of five known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors. 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. The sums of the first five non-primes greater than zero and the first five prime numbers 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 formula_16−1(formula_16 − formula_3 ) with a formula_19 of formula_1, by the Euclid–Euler theorem. 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.
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. 
In figurate numbers, 5 is a pentagonal number, with the sequence of pentagonal numbers starting: 1, 5, 12, 22, 35, ...
The factorial of five, or formula_1! = formula_22, is also 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. 35, which is the fourth or fifth pentagonal and tetrahedral number, is equal to the sum of the first five triangular numbers: 1, 3, 6, 10, 15.
5 is the value of the central cell of the only non-trivial normal magic square, also called the "Lo Shu" square. Its formula_23 x formula_23 array of squares has a magic constant formula_25 of formula_26, where the sums of its rows, columns, and diagonals are all equal to fifteen. 5 is also the value of the central cell the only non-trivial order-3 normal magic hexagon that is made of nineteen cells.
Polynomial equations of degree and below can be solved with radicals, while quintic equations of degree 5, and higher, cannot generally be so solved. This is the Abel–Ruffini theorem. This is related to the fact that the symmetric group formula_27 is a solvable group for formula_8 ⩽ formula_29, and not for formula_8 ⩾ formula_1.
In the Collatz problem, 5 requires five steps to reach 1 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). 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. In the analogous problem, 5 requires five steps to return cyclically to 5, in this instance by multiplying terms by three and "subtracting" 1 if the terms are odd, and also halving if even: {5 ➙ 14 ➙ 7 ➙ 20 ➙ 10 ➙ 5 ➙ ...}. This is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...) while also generating a {14 ➙ ... ➙ 14} cycle that appears in other trajectories (alongside those generated by the other numbers in the path cycle of 5).
There are five countably infinite Ramsey classes of permutations, where the age of each countable homogeneous permutation forms an individual Ramsey class formula_32 of objects such that, for each natural number formula_33 and each choice of objects formula_34, there is no object formula_35 where in any formula_33-coloring of all subobjects of formula_37 isomorphic to formula_38 there is a monochromatic subobject isomorphic to formula_39. In general, the Fraïssé limit of a class formula_32 of finite relational structure is the age of a countable homogeneous relational structure formula_41 iff five conditions hold for formula_32: 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.
Euler's identity, formula_43+ formula_3 = formula_45, contains five essential numbers used widely in mathematics: Archimedes' constant formula_46, Euler's number formula_47, the imaginary number formula_48, unity formula_3, and zero formula_45, which makes this formula a renown example of beauty in mathematics.
In 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, formula_51. 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 }. 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. Generally, star polytopes that are regular only exist in dimensions formula_52 ⩽ formula_8 &lt; formula_1, and can be constructed using five Miller rules for stellating polyhedra or higher-dimensional polytopes.
In graph theory, all graphs with 4 or fewer vertices are planar, however, there is a graph with 5 vertices that is not: "K"5, the complete graph with 5 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 "K"5, or the complete bipartite utility graph "K"3,3. 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. The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles. The automorphism group of the Petersen graph is the symmetric group formula_55 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. 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 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 are generated from combinations of only five regular polygons: the triangle, square, hexagon, octagon, and the dodecagon. The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.
Five points are needed to determine a conic section, in the same way that two points are needed to determine a line. A Veronese surface in the projective plane formula_56 of a conic generalizes a linear condition for a point to be contained inside a conic.
There are five Platonic solids in three-dimensional space: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. 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:
The pentatope, or 5-cell, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry formula_59 of order 120 = 5! and formula_55 group structure. Made of five tetrahedra, its Petrie polygon is a regular pentagon and its orthographic projection is equivalent to the complete graph "K"5. 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.
Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora: formula_62, formula_63, formula_64, formula_65, and formula_66, 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 25 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 formula_67 hexadecachoric or formula_68 icositetrachoric symmetry do not exist in dimensions formula_8 ⩾ formula_1; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have formula_67 and formula_68 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. Only two regular projective polytopes exist in each higher dimensional space.
The 5-simplex or "hexateron" is the five-dimensional analogue of the 5-cell, or 4-simplex. It has Coxeter group formula_73 as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group formula_74, 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 formula_75 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. There are exclusively twelve complex aperiotopes in formula_76 complex spaces of dimensions formula_8 ⩾ formula_1, with fifteen in formula_79 and sixteen in formula_80; alongside complex polytopes in formula_81 and higher under simplex, hypercubic and orthoplex groups, the latter with van Oss polytopes.
There are five exceptional Lie algebras: formula_82, formula_83, formula_84, formula_85, and formula_86. The smallest of these, formula_82, 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. formula_86 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. This sphere packing formula_89 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. There are specifically five solvable groups that are excluded from finite simple groups of Lie type.
The five Mathieu groups constitute the in the happy family of sporadic groups. These are also the first five sporadic groups to have been described, defined as formula_90 multiply transitive permutation groups on formula_8 objects, with formula_8 ∈ {11, 12, 22, 23, 24}. In particular, formula_93, 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 formula_8 elements. Of precisely five different conjugacy classes of maximal subgroups of formula_93, one is the almost simple symmetric group formula_96 (of order 5!), and another is formula_97, also almost simple, that functions as a point stabilizer which has formula_1 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 formula_93 is sharply 4-transitive, formula_100 is sharply 5-transitive and formula_101 is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups. formula_102 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. All Mathieu groups are subgroups of formula_101, which under the Witt design formula_104 of Steiner system S(5, 8, 24) emerges a construction of the extended binary Golay code formula_105 that has formula_101 as its automorphism group. formula_104 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. 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 of seven sporadic groups that are subquotients of the automorphism of the Leech lattice, Conway group formula_108.
There are five non-supersingular primes: 37, 43, 53, 61, and 67, all smaller than the largest of fifteen supersingular prime divisors of the friendly giant, 71.
List of basic calculations.
In decimal.
5 is the only prime number to end in the digit 5 in decimal because all other numbers written with a 5 in the ones place are multiples of five, which makes it a 1-automorphic number.
All multiples of 5 will end in either 5 or , and vulgar fractions with 5 or 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 formula_8 raised to the fifth power always ends in the same digit as formula_8.
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 different 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 different ways, producing from that were more similar to the digits 4 or 3 than to 5. 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, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa.
Miscellaneous fields.
Five can refer to: