2005 In Music

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 attained significance throughout history in part because typical humans have five digits on each hand.

In mathematics

$5$ 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 a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial, and an Eisenstein prime with no imaginary part and real part of the form $3p$ − $1$. 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 $2^$+ $1$, and more generally the second Sierpiński number of the first kind, $n^n$+ $1$. There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537. The sum of the first 3 Fermat primes, 3, 5 and 17, yields 25 or 5², while 257 is the 55th prime number. Combinations from these 5 Fermat primes generate 31 polygons with an odd number of sides that can be construncted 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 $n$-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 $2^n$ − $1$, which yields $31$: 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 $M_$ = 2^{23058...93951} − 1 too large to compute with current computers. In a related sequence, the first 5 terms in the sequence of Catalan–Mersenne numbers $M_$ 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.

Every odd number greater than $1$ is the sum of at most five prime numbers, and every odd number greater than $5$ is conjectured to be expressible as the sum of three prime numbers. Helfgtott 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.

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 7th triangular number and like $6$ a perfect number, which also includes $496$, the 31st triangular number and perfect number of the form $2^$⁻¹($2^$ − $1$) with a $p$ of $5$, by the Euclid–Euler theorem.

There are a total of five known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors. A sixth unitary number, if discovered, would have at least nine odd prime factors.

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, ...
* 5 is a centered tetrahedral number: 1, 5, 15, 35, 69, ... 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, ... The sum of the first four members is 50 while the fifth indexed member in the sequence is 55.
* 5 is a centered square number: 1, 5, 13, 25, 41, ... 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.

The factorial of five, or $5$ ! = $120$, 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. 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 $3$ x $3$ array of squares has a magic constant $M$ of $15$, 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 $\mathrm_$ is a solvable group for ''n'' ⩽ 4 and not solvable for ''n'' ⩾ 5.

Euler's identity, $e^$+ $1$ = $0$, contains five essential numbers used widely in mathematics: Archimedes' constant $\pi$, Euler's number $e$, the imaginary number $i$, unity $1$, and zero $0$, 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, $\varphi$. 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 . A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges. Generally, star polytopes that are regular only exist in dimensions 2 ⩽ $n$ < 5.

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.Royle, G
"Cubic Symmetric Graphs (The Foster Census)."
The automorphism group of the Petersen graph is the symmetric group $\mathrm_$ 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. 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 $\mathbb^5$ of a conic generalizes a linear condition for a point to be contained inside a conic.

There are $5$ 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 $5$:

☆ Regular polyhedron compounds: the stella octangula, compound of five tetrahedra, compound of five cubes, compound of five octahedra, and compound of ten tetrahedra. Icosahedral symmetry $\mathrm I_$ is isomorphic to the alternating group on 5 letters $\mathrm A_$ of order 120, realized by actions on these uniform polyhedron compounds.

☆ Space-filling convex polyhedra: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium. While the cube is the only Platonic solid that can tessellate space on its own, the truncated octahedron and the gyrobifastigium are the only Archimedean and Johnson solids, respectively, that can also tessellate space with their own copies.

☆ Cell-transitive parallelohedra: any parallelepiped, as well as the rhombic dodecahedron and elongated dodecahedron, and the hexagonal prism and truncated octahedron. The cube is a special case of a parallelepiped, with the rhombic dodecahedron the only Catalan solid to tessellate space on its own.

☆ Regular abstract polyhedra, which include the excavated dodecahedron and the dodecadodecahedron. 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.

The 5-cell, or pentatope, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry $\mathrm_$ of order 120 = 5 ! and $\mathrm_$ 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.

*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 and made of fifty tetrahedra, ten pentagrammic crossed antiprisms, ten pentagonal antiprisms, and fifty vertices.

*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.

*The abstract four-dimensional 57-cell is made of fifty-seven hemi-icosahedral cells, in-which five surround each edge. The 11-cell, another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven hemi-dodecahedral cells each with fifteen dodecahedra. The skeleton of the hemi-dodecahedron is the Petersen graph.

Overall, the fourth dimension contains five Weyl groups that form a finite number of uniform polychora: $\mathrm A_$, $\mathrm B_$, $\mathrm D_$, $\mathrm F_$, and $\mathrm H_$, with four of these Coxeter groups capable of generating the same finite forms without $\mathrm D_$; 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 Eucledian 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 three-dimensional icosahedral symmetry $\mathrm_$ or four-dimensional $\mathrm_$ symmetry do not exist in dimensions ''n'' ⩾ 5; however, there is the uniform prismatic group $\mathrm_$ × $\mathrm_$ in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have $\mathrm_$ symmetry.

The 5-simplex is the five-dimensional analogue of the 5-cell, or 4-simplex; the fifth iteration of $n$-simplexes in any $n$ dimensions. The 5-simplex has the Coxeter group $\mathrm_$ as its symmetry group, of order 720 = 6 !, whose group structure is represented by the symmetric group $\mathrm_$, 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 $\mathrm B_$ 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. There are exclusively twelve complex aperiotopes in $\mathbb^n$ complex spaces of dimensions $n$ ⩾ $5$, with fifteen in $\mathbb^4$ and sixteen in $\mathbb^3$; alongside complex polytopes in $\mathbb^5$ and higher under simplex, hypercubic and orthoplex groups, the latter with van Oss polytopes.

There are five exceptional Lie groups: $\mathfrak_2$, $\mathfrak_4$, $\mathfrak_6$, $\mathfrak_7$, and $\mathfrak_8$. The smallest of these, $\mathfrak_2$, can be represented in five-dimensional complex space and projected in the same number of dimensions as a ball rolling on top of another ball, whose motion is described in two-dimensional space. $\mathfrak_8$, the largest of all five exceptional groups, also contains the other four as subgroups and is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the 600-cell. There are also five solvable groups that are excluded from finite simple groups of Lie type.

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 $\mathrm_$ multiply transitive permutation groups on $n$ objects, with $n$ ∈ . In particular, $\mathrm_$, 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 ''n'' elements. Of precisely five different conjugacy classes of maximal subgroups of $\mathrm_$, one is the almost simple symmetric group $\mathrm_5$ (of order 5 !), and another is $\mathrm_$, also almost simple, that functions as a point stabilizer which has $5$ 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 $\mathrm_$ is sharply 4-transitive, $\mathrm_$ is sharply 5-transitive and $\mathrm_$ is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups. $\mathrm_$ 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 $\mathrm_$, which under the Witt design $\mathrm_$ of Steiner system S(5, 8, 24) emerges a construction of the extended binary Golay code $\mathrm_$ that has $\mathrm_$ as its automorphism group. $\mathrm_$ 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 second generation of seven sporadic groups that are subquotients of the automorphism of the Leech lattice, Conway group $\mathrm_$.

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 $n$ raised to the fifth power always ends in the same digit as $n$.

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.

Science

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

Astronomy

* Messier object M5, a magnitude 7.0 globular cluster in the constellation Serpens.
*The New General Cataloguebr>object
NGC 5, a magnitude 13 spiral galaxy

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