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'''5''' ('''five''') is a [[number]], [[numeral (linguistics)|numeral]] and [[numerical digit|digit]]. It is the [[natural number]], and [[cardinal number]], following [[4]] and preceding [[6]], and is a [[prime number]].
'''5''' ('''five''') is a [[number]], [[numeral (linguistics)|numeral]] and [[numerical digit|digit]]. It is the [[natural number]], and [[cardinal number]], following 4 and preceding [[6]], and is a [[prime number]].


Humans, and many other animals, have 5 [[Digit (anatomy)|digits]] on their [[Limb (anatomy)|limbs]].
Humans, and many other animals, have 5 [[Digit (anatomy)|digits]] on their [[Limb (anatomy)|limbs]].
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[[File:Pythagoras' Special Triples.svg|right|upright=0.65|thumb|The first [[Pythagorean triple]] ]]
[[File:Pythagoras' Special Triples.svg|right|upright=0.65|thumb|The first [[Pythagorean triple]] ]]


Five is the second [[Fermat prime]], the third [[Mersenne prime]] exponent, as well as a [[Fibonacci number]]. 5 is the first [[congruent number]], as well as the length of the [[hypotenuse]] of the smallest integer-sided [[right triangle]], making part of the smallest [[Pythagorean triple]] ([[3]], [[4]], 5).<ref>{{Cite OEIS|1=A003273 |2=Congruent numbers|access-date=2016-06-01}}</ref>
5 is a [[Fermat prime]], a [[Mersenne prime]] exponent, as well as a [[Fibonacci number]]. 5 is the first [[congruent number]], as well as the length of the [[hypotenuse]] of the smallest integer-sided [[right triangle]], making part of the smallest [[Pythagorean triple]] ([[3]], [[4]], 5).<ref>{{Cite OEIS|1=A003273 |2=Congruent numbers|access-date=2016-06-01}}</ref>

5 is the first [[safe prime]],<ref>{{Cite OEIS|A005385|Safe primes p: (p-1)/2 is also prime|access-date=2023-02-14}}</ref> and the first [[good prime]].<ref>{{Cite OEIS|A028388|Good primes|access-date=2016-06-01}}</ref> 11 forms the first pair of [[sexy prime]]s with 5.<ref>{{Cite OEIS|A023201|Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)|access-date=2023-01-14}}</ref> 5 is the second [[Fermat number|Fermat prime]], of a total of five known Fermat primes.<ref>{{Cite OEIS|A019434|Fermat primes|access-date=2022-07-21}}</ref> 5 is also the first of three known [[Wilson prime]]s (5, 13, 563).<ref name="WilsP">{{Cite OEIS|A007540|Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).|access-date=2023-09-06}}</ref>


=== Geometry ===
=== Geometry ===
A shape with five sides is called a [[pentagon]]. The pentagon is the first [[regular polygon]] that does not [[Tessellation|tile]] the [[Plane (geometry)|plane]] with copies of itself. It is the largest [[Face (geometry)|face]] any of the five regular three-dimensional regular [[Platonic solid]] can have.
A shape with five sides is called a [[pentagon]]. The pentagon is the first [[regular polygon]] that does not [[Tessellation|tile]] the [[Plane (geometry)|plane]] with copies of itself. It is the largest [[Face (geometry)|face]] any of the five regular three-dimensional regular [[Platonic solid]] can have.


A [[conic]] is determined using [[Five points determine a conic|five points in the same way]] that two points are needed to determine a [[Line (geometry)|line]].<ref name="conic">{{Cite journal |first=A. C. |last=Dixon |author-link=Alfred Cardew Dixon |title=The Conic through Five Given Points |journal=The Mathematical Gazette |volume=4 |number=70 |date=March 1908 |pages=228–230 |publisher=The Mathematical Association |jstor=3605147 |doi=10.2307/3605147 |s2cid=125356690 |url=https://zenodo.org/record/2014634 }}</ref>A [[pentagram]], or five-pointed [[Polygram (geometry)|polygram]], is a [[star polygon]] constructed by connecting some non-adjacent of a [[regular pentagon]] as [[Star polygon#Regular star polygon|self-intersecting edges]].<ref>{{Cite OEIS|A307681|Difference between the number of sides and the number of diagonals of a convex n-gon.}}</ref>
A [[conic]] is determined using [[Five points determine a conic|five points in the same way]] that two points are needed to determine a [[Line (geometry)|line]].<ref name="conic">{{Cite journal |first=A. C. |last=Dixon |author-link=Alfred Cardew Dixon |title=The Conic through Five Given Points |journal=The Mathematical Gazette |volume=4 |number=70 |date=March 1908 |pages=228–230 |publisher=The Mathematical Association |jstor=3605147 |doi=10.2307/3605147 |s2cid=125356690 |url=https://zenodo.org/record/2014634 }}</ref> A [[pentagram]], or five-pointed [[Polygram (geometry)|polygram]], is a [[star polygon]] constructed by connecting some non-adjacent of a [[regular pentagon]] as [[Star polygon#Regular star polygon|self-intersecting edges]].<ref>{{Cite OEIS|A307681|Difference between the number of sides and the number of diagonals of a convex n-gon.}}</ref> The internal geometry of the pentagon and pentagram (represented by its [[Schläfli symbol]] {{math|1={5/2<nowiki>}</nowiki> }}) appears prominently in [[Penrose tilings]]. Pentagrams are [[Facet (geometry)|facet]]s inside [[Kepler–Poinsot polyhedron|Kepler–Poinsot star polyhedra]] and [[Regular 4-polytope#Regular star (Schläfli–Hess) 4-polytopes|Schläfli–Hess star polychora]].


There are five regular [[Platonic solid]]s the [[tetrahedron]], the [[cube]], the [[octahedron]], the [[dodecahedron]], and the [[icosahedron]].<ref>Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 61</ref>
5 is the first [[safe prime]]<ref>{{Cite OEIS|A005385|Safe primes p: (p-1)/2 is also prime|access-date=2023-02-14}}</ref> where <math>(p - 1)/2</math> for a prime <math>p</math> is also prime ([[2]]), and the first [[good prime]], since it is the first prime number whose square ([[25 (number)|25]]) is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes (i.e., [[3]] × [[7]] = [[21 (number)|21]] and [[11 (number)|11]] × [[2]] = [[22 (number)|22]] are less than 25).<ref>{{Cite OEIS|A028388|Good primes|access-date=2016-06-01}}</ref> 11, the fifth prime number, is the next good prime, that also forms the first pair of [[sexy prime]]s with 5.<ref>{{Cite OEIS|A023201|Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)|access-date=2023-01-14}}</ref> 5 is the second [[Fermat number|Fermat prime]] of the form <math>2^{2^{n}} + 1</math>, of a total of five known Fermat primes.<ref>{{Cite OEIS|A019434|Fermat primes|access-date=2022-07-21}}</ref>


The [[Hadwiger–Nelson problem|chromatic number]] of the [[Plane (geometry)|plane]] is the minimum number of [[Graph coloring|colors]] required to color the plane such that no pair of points at a distance of 1 has the same color.<ref>{{Cite journal |last=de Grey |first=Aubrey D.N.J. |author-link=Aubrey de Grey |year=2018 |title=The Chromatic Number of the Plane is At Least 5 |journal=[[Geombinatorics]] |volume=28 |pages=5–18 |arxiv=1804.02385 |mr=3820926 |s2cid=119273214}}</ref> Five is a lower depending for the chromatic number of the plane, but this may depend on the choice of [[Axiom of choice|set-theoretical axioms]]:<ref>{{Cite journal |last1=Exoo |first1=Geoffrey |last2=Ismailescu |first2=Dan |year=2020 |title=The Chromatic Number of the Plane is At Least 5: A New Proof |journal=[[Discrete & Computational Geometry]] |location=New York, NY |publisher=[[Springer Science+Business Media|Springer]] |volume=64 |pages=216–226 |arxiv=1805.00157 |doi=10.1007/s00454-019-00058-1 |mr=4110534 |s2cid=119266055 |zbl=1445.05040}}</ref>
The internal geometry of the pentagon and pentagram (represented by its [[Schläfli symbol]] {{math|1={5/2<nowiki>}</nowiki> }}) appears prominently in [[Penrose tilings]], and they are [[Facet (geometry)|facet]]s inside [[Kepler–Poinsot polyhedron|Kepler–Poinsot star polyhedra]] and [[Regular 4-polytope#Regular star (Schläfli–Hess) 4-polytopes|Schläfli–Hess star polychora]]. A similar figure to the pentagram is a [[Five-pointed star|five-pointed]] [[Simple polygon|simple]] [[isotoxal]] star ☆ without self-intersecting edges, often found inside Islamic [[Girih tiles]] (there are five different rudimentary types).<ref>{{Cite journal |last=Sarhangi |first=Reza |year=2012 |title=Interlocking Star Polygons in Persian Architecture: The Special Case of the Decagram in Mosaic Designs |url=https://link.springer.com/content/pdf/10.1007/s00004-012-0117-5.pdf |journal=Nexus Network Journal |volume=14 |issue=2 |page=350 |doi=10.1007/s00004-012-0117-5 |s2cid=124558613 |doi-access=free}}</ref>


The plane contains a total of five [[Bravais lattice]]s, or arrays of [[Point (geometry)|points]] defined by discrete [[Translation (geometry)|translation]] operations. [[Euclidean tilings by convex regular polygons|Uniform tilings]] of the plane, are generated from combinations of only five regular polygons.<ref>{{Cite journal |last1=Grünbaum |first1=Branko |author-link=Branko Grünbaum |last2=Shepard |first2=Geoffrey |author-link2=G.C. Shephard |date=November 1977 |title=Tilings by Regular Polygons |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=[[Mathematics Magazine]] |publisher=Taylor & Francis, Ltd. |volume=50 |issue=5 |pages=227–236 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006}}</ref>
=== Algebra ===
5 is the value of the central [[Magic square#Properties of magic squares|cell]] of the first non-trivial [[magic square|normal magic square]], called the [[Luoshu Square|''Luoshu'' square]]. 5 is also the first of three known [[Wilson prime]]s (5, 13, 563),<ref name="WilsP">{{Cite OEIS|A007540|Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).|access-date=2023-09-06}}</ref> where the square of a prime <math>p^2</math> [[divisor|divides]] <math>(p-1)!+1.</math> As a consequence of [[Fermat's little theorem]] and [[Euler's criterion]], all [[Square number|squares]] are [[Modular arithmetic#Congruence|congruent]] to [[0]], [[1]], 4 (or [[−1]]) [[Modulo operation|modulo]] 5.<ref>{{Cite journal |last=Sellers |first=James A. |year=2013 |title=An unexpected congruence modulo 5 for 4-colored generalized Frobenius partitions |journal=J. Indian Math. Soc. |series=New Series |location=Pune, IMD |publisher=[[Indian Mathematical Society]] |issue=Special Issue |page=99 |arxiv=1302.5708 |bibcode=2013arXiv1302.5708S |mr=157339 |s2cid=116931082 |zbl=1290.05015}}</ref> All [[integer]]s <math>n \ge 34</math> can be expressed as the sum of five non-zero [[Square number|squares]].<ref>{{Cite book |last1=Niven |first1=Ivan |author1-link=Ivan M. Niven |title=An Introduction to the Theory of Numbers |last2=Zuckerman |first2=Herbert S. |last3=Montgomery |first3=Hugh L. |author3-link=Hugh Lowell Montgomery |publisher=[[Wiley (publisher)|John Wiley]] |year=1980 |isbn=978-0-19-853171-5 |edition=5th |location=New York, NY |pages=144, 145}}</ref><ref>{{Cite OEIS|A047701|All positive numbers that are not the sum of 5 nonzero squares.|access-date=2023-09-20}}


=== Higher Dimensional Geometry ===
: Only twelve integers up to [[33 (number)|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.</ref> There are five countably infinite [[Ramsey class]]es of [[permutation]]s.<ref name="ramsey">{{Cite journal |last1=Böttcher |first1=Julia |author1-link=Julia Böttcher |last2=Foniok |first2=Jan |year=2013 |title=Ramsey Properties of Permutations |url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p2 |journal=The Electronic Journal of Combinatorics |volume=20 |issue=1 |page=P2 |arxiv=1103.5686v2 |doi=10.37236/2978 |s2cid=17184541 |zbl=1267.05284}}</ref>{{rp|p.4}}
A [[5-cell|hypertetrahedron]], or 5-cell, is the 4 dimensional analogue of the [[tetrahedron]]. It has five vertices. Its orthographic projection is [[Homomorphism|homomorphic]] to the group ''K''<sub>5.</sub><ref name="RegPoly3">{{Cite book |author=H. S. M. Coxeter |author-link=Harold Scott MacDonald Coxeter |title=[[Regular Polytopes (book)|Regular Polytopes]] |publisher=[[Dover Publications, Inc.]] |year=1973 |isbn=978-0-486-61480-9 |edition=3rd |location=New York |pages=1–368}}</ref>{{rp|p.120}}


There are five fundamental [[Uniform 4-polytope#Convex uniform 4-polytopes|mirror symmetry point group families in 4-dimensions]]. There are also 5 [[Coxeter-Dynkin diagram#Compact|compact hyperbolic Coxeter groups]], or [[Uniform 4-polytope#Prismatic uniform 4-polytopes|4-prisms]], of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.<ref>{{Cite book |last1=McMullen |first1=Peter |author1-link=Peter McMullen |url=https://archive.org/details/abstractregularp0000mcmu |title=Abstract Regular Polytopes |last2=Schulte |first2=Egon |author2-link=Egon Schulte |publisher=Cambridge University Press |year=2002 |isbn=0-521-81496-0 |series=Encyclopedia of Mathematics and its Applications |volume=92 |location=Cambridge |pages=162–164 |doi=10.1017/CBO9780511546686 |mr=1965665 |url-access=registration |s2cid=115688843}}</ref>
Five is [[conjecture]]d to be the only [[Parity (mathematics)|odd]], [[untouchable number]]; if this is the case, then five will be the only odd prime number that is not the base of an [[Aliquot sequence|aliquot]] tree.<ref>{{Cite journal |last1=Pomerance |first1=Carl |last2=Yang |first2=Hee-Sung |date=14 June 2012 |title=On Untouchable Numbers and Related Problems |url=https://math.dartmouth.edu/~carlp/uupaper3.pdf |publisher=[[Dartmouth College]] |page=1 |s2cid=30344483 |website=math.dartmouth.edu}} 2010 Mathematics Subject Classification. 11A25, 11Y70, 11Y16.</ref>

Every odd number greater than five is conjectured to be expressible as the sum of three prime numbers; [[Harald Andres Helfgott|Helfgott]] has provided a proof of this<ref>{{Cite book |last=Helfgott |first=Harald Andres |title=Seoul [[International Congress of Mathematicians]] Proceedings |date=2014 |publisher=Kyung Moon SA |isbn=978-89-6105-805-6 |editor-last=Jang |editor-first=Sun Young |volume=2 |location=Seoul, KOR |pages=391–418 |chapter=The ternary Goldbach problem |oclc=913564239 |chapter-url=https://www.imj-prg.fr/wp-content/uploads/2020/prix/helfgott2014.pdf}}</ref> (also known as the [[Goldbach's weak conjecture|odd Goldbach conjecture]]) that is already widely acknowledged by mathematicians as it still undergoes [[peer-review]]. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).<ref>{{Cite journal |last=Tao |first=Terence |date=March 2014 |title=Every odd number greater than 1 has a representation is the sum of at most five primes |url=https://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02733-0/S0025-5718-2013-02733-0.pdf |journal=Mathematics of Computation |volume=83 |pages=997–1038 |doi=10.1090/S0025-5718-2013-02733-0 |mr=3143702 |s2cid=2618958 |number=286}}</ref>{{Unsolved|mathematics|Is 5 the only odd, untouchable number?}}
[[File:Magic Square Lo Shu.svg|upright=0.55|right|thumb|The smallest non-trivial [[magic square]]]][[File:Five Pointed Star Lined.svg|left|135px]]

=== Graph theory, and planar geometry ===
In [[graph theory]], all [[Graph theory|graphs]] with four or fewer vertices are [[Planar graph|planar]], however, there is a graph with five vertices that is not: ''K''<sub>5</sub>, the [[complete graph]] with five vertices,. By [[Kuratowski's theorem]], a finite graph is planar [[iff|if and only if]] it does not contain a subgraph that is a subdivision of ''K''<sub>5</sub>, or ''K''<sub>3,3</sub>, the [[utility graph]].<ref>{{Cite journal |last=Burnstein |first=Michael|title=Kuratowski-Pontrjagin theorem on planar graphs |journal=[[Journal of Combinatorial Theory]] | series=Series B |volume=24 |issue=2 |year=1978 |pages=228–232 |doi= 10.1016/0095-8956(78)90024-2 |doi-access=free }}</ref>

The [[Hadwiger–Nelson problem|chromatic number]] of the [[Plane (geometry)|plane]] is the minimum number of [[Graph coloring|colors]] required to color the plane such that no pair of points at a distance of 1 has the same color.<ref>{{Cite journal |last=de Grey |first=Aubrey D.N.J. |author-link=Aubrey de Grey |title=The Chromatic Number of the Plane is At Least 5 |journal=[[Geombinatorics]] |volume=28 |pages=5–18 |year=2018 |arxiv=1804.02385 |mr=3820926 |s2cid=119273214 }}</ref> Five is a lower depending for the chromatic number of the plane, but this may depend on the choice of [[Axiom of choice|set-theoretical axioms]]:<ref>{{Cite journal |last1=Exoo |first1=Geoffrey |last2=Ismailescu |first2=Dan |title=The Chromatic Number of the Plane is At Least 5: A New Proof |journal=[[Discrete & Computational Geometry]] |volume=64 |pages=216–226 |publisher=[[Springer Science+Business Media|Springer]] |location=New York, NY |year=2020 |doi=10.1007/s00454-019-00058-1 |arxiv=1805.00157 |mr=4110534 |s2cid=119266055 |zbl=1445.05040 }}</ref>

The plane contains a total of five [[Bravais lattice]]s, or arrays of [[Point (geometry)|points]] defined by discrete [[Translation (geometry)|translation]] operations: [[Hexagonal lattice|hexagonal]], [[Oblique lattice|oblique]], [[Rectangular lattice|rectangular]], [[Rectangular lattice#Bravais lattices|centered rectangular]], and [[Square lattice|square]] lattices. [[Euclidean tilings by convex regular polygons|Uniform tilings]] of the plane, furthermore, are generated from combinations of only five regular polygons: the [[triangle]], [[square]], [[hexagon]], [[octagon]], and the [[dodecagon]].<ref>{{Cite journal |first1=Branko |last1=Grünbaum |author-link=Branko Grünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=[[Mathematics Magazine]] |volume=50 |issue=5 |publisher=Taylor & Francis, Ltd.|pages=227–236 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 }}</ref>

=== Polyhedral geometry ===

[[File:De divina proportione - Illustration 13, crop.jpg|upright |right|thumb|Illustration by [[Leonardo da Vinci]] of a [[regular dodecahedron]], from [[Luca Pacioli]]'s ''[[Divina proportione]]'']]

There are five [[Platonic solid]]s in [[three-dimensional space]] that are [[Regular polyhedron|regular]]: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.<ref>Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 61</ref> The [[Regular dodecahedron|dodecahedron]] in particular contains [[pentagonal]] faces, while the [[Regular icosahedron|icosahedron]], its [[Dual polytope|dual polyhedron]], has a [[vertex figure]] that is a regular pentagon. These five regular solids are responsible for generating thirteen figures that classify as [[Semiregular polyhedron|semi-regular]], which are called the [[Archimedean solid]]s. There are also five:

{{Bullet list
|[[Uniform polyhedron compound|Regular polyhedron compounds]]: the [[compound of five tetrahedra]], compound of ten tetrahedra, compound of five cubes, compound of five octahedra, and [[stella octangula]].<ref name="JSkill">{{cite journal |first=John |last=Skilling |title=Uniform Compounds of Uniform Polyhedra |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=79 |pages=447–457 |year=1976 |issue=3 |doi=10.1017/S0305004100052440 |bibcode=1976MPCPS..79..447S |mr=0397554|s2cid=123279687 }}</ref> [[Icosahedral symmetry]] <math>\mathrm I_{h}</math> is [[isomorphic]] to the [[alternating group]] on five letters <math>\mathrm A_{5}</math> of order [[120 (number)|120]], realized by actions on these uniform polyhedron compounds (aside from the regular compound of two tetrahedra). All fifteen [[mirror plane]]s of <math>\mathrm I_{h}</math> pass through the edges of a regular [[Spherical polyhedron|spherical]] [[compound of five octahedra]], whose sets of three orthogonal [[great circle]]s use five colors.{{efn|1=[[File:Spherical disdyakis triacontahedron as compound of five octahedra.png |165px]]}}<ref>{{Cite conference |editor-last=Sarhangi |editor-first=Reza |author-last=Hart |author-first=George W. |author-link= George W. Hart |title=Icosahedral Constructions |url=http://t.archive.bridgesmathart.org/1998/bridges1998-195.pdf |book-title=Bridges: Mathematical Connections in Art, Music, and Science |publisher=[[The Bridges Organization]] |location=Winfield, Kansas |year=1998 |page=196 |isbn=978-0-9665201-0-1 |oclc=59580549 |s2cid=202679388 }}</ref><ref>{{Cite web |last=Hart |first=George W. |author-link= George W. Hart |title=Symmetry Planes |website=Virtual Polyhedra (The Encyclopedia of Polyhedra) |url=https://www.georgehart.com/virtual-polyhedra/symmetry_planes.html |access-date=2023-09-27 }}
:"They can be colored as five sets of three mutually orthogonal planes" where the "fifteen planes divide the sphere into 120 [[Schwarz triangle|Möbius triangle]]s."</ref>
|[[Honeycomb (geometry)|Space-filling]] [[convex polyhedron|convex polyhedra]] with regular faces: the triangular prism, [[hexagonal prism]], cube, truncated octahedron, and [[gyrobifastigium]].<ref>{{Cite book |title=The Six-Cornered Snowflake |first=Johannes |last=Kepler |author-link=Johannes Kepler |publisher=Paul Dry Books |year=2010 |isbn=978-1-58988-285-0 |at=Footnote 18, [https://books.google.com/books?id=yE8yTUFWLXgC&pg=PA146 p.&nbsp;146] }}</ref> 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 solid]]s, respectively, that can tessellate space with their own copies.
|[[Isohedral figure#Related terms|Cell-transitive]] [[Parallelohedron|parallelohedra]]: any [[parallelepiped]], as well as the [[rhombic dodecahedron]], the [[elongated dodecahedron]], the hexagonal prism and the truncated octahedron.<ref>{{cite book|last=Alexandrov|first=A. D.|author-link=Aleksandr Danilovich Aleksandrov|contribution=8.1 Parallelohedra|pages=349–359|publisher=Springer|title=Convex Polyhedra|title-link=Convex Polyhedra (book)|year=2005}}</ref> The cube is a special case of a parallelepiped, and the rhombic dodecahedron (with five [[stellation]]s per [[Stellation#Miller's rules|Miller's rules]]) is the only [[Catalan solid]] to tessellate space on its own.<ref>{{Cite web |url=https://www.software3d.com/Enumerate.php |last=Webb |first=Robert |title=Enumeration of Stellations |website=www.software3d.com |access-date=2023-01-12 |archive-url=https://archive.today/20221126015207/https://www.software3d.com/Enumerate.php |archive-date=2022-11-26 }}</ref>
|[[List of regular polytopes and compounds#Abstract polytopes|Regular abstract polyhedra]], which include the [[excavated dodecahedron]] and the [[dodecadodecahedron]].<ref>{{cite journal |last=Wills |first=J. M. |title=The combinatorially regular polyhedra of index 2 |journal=Aequationes Mathematicae |volume=34 |year=1987 |issue=2–3 |pages=206–220 |doi=10.1007/BF01830672 |s2cid=121281276 }}</ref> They have combinatorial symmetries transitive on [[List of regular polytopes and compounds#Abstract polytopes|flags]] of their elements, with [[Topology|topologies]] equivalent to that of [[toroid]]s and the ability to tile the [[hyperbolic plane]].
}}

Moreover, there are also precisely ''five'' [[uniform prism]]s and [[antiprism]]s that contain pentagons or pentagrams as faces — the [[pentagonal prism]] and [[Pentagonal antiprism|antiprism]], and the [[pentagrammic prism]], [[Pentagrammic antiprism|antiprism]], and [[Pentagrammic crossed antiprism|crossed-antiprism]].<ref name="HarEl">{{Cite journal |last=Har'El |first=Zvi |url=http://harel.org.il/zvi/docs/uniform.pdf |title=Uniform Solution for Uniform Polyhedra |journal=[[Geometriae Dedicata]] |volume=47 |pages=57–110 |publisher=[[Springer Publishing]] |location=Netherlands |year=1993 |doi=10.1007/BF01263494 |mr=1230107 |zbl=0784.51020 |s2cid=120995279 }}
:"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 triangle]]s."<br />Appendix II: Uniform Polyhedra.</ref>

=== Four-dimensional space ===


[[File:Schlegel wireframe 5-cell.png|upright|left|thumb|The four-dimensional [[5-cell]] is the simplest regular [[polychoron]].]]
[[File:Schlegel wireframe 5-cell.png|upright|left|thumb|The four-dimensional [[5-cell]] is the simplest regular [[polychoron]].]]


=== Algebra ===
The [[5-cell|pentatope]], or 5-cell, is the self-dual fourth-dimensional analogue of the [[tetrahedron]], with [[Coxeter group]] symmetry <math>\mathrm{A}_{4}</math> of order [[120 (number)|120]] = 5[[Factorial|!]] and <math>\mathrm{S}_{5}</math> [[Mathematical structure|group structure]]. Made of five tetrahedra, its [[Petrie polygon]] is a [[regular pentagon]] and its [[orthographic projection]] is equivalent to the [[complete graph]] ''K''<sub>5</sub>. It is one of six [[Uniform 4-polytope|regular 4-polytopes]], made of thirty-one [[Simplex#Elements|elements]]: five [[Vertex (geometry)|vertices]], ten [[Edge (geometry)|edges]], ten [[Face (geometry)|faces]], five [[Regular tetrahedron|tetrahedral cells]] and one [[Face (geometry)#Facet or (n − 1)-face|4-face]].<ref name="RegPoly3">{{Cite book |author=H. S. M. Coxeter |author-link=Harold Scott MacDonald Coxeter |title=[[Regular Polytopes (book)|Regular Polytopes]] |publisher=[[Dover Publications, Inc.]] |edition=3rd |year=1973 |location=New York |pages=1–368 |isbn=978-0-486-61480-9 }}</ref>{{rp|p.120}}
[[File:Magic Square Lo Shu.svg|upright=0.55|right|thumb|The smallest non-trivial [[magic square]]]]5 is the value of the central [[Magic square#Properties of magic squares|cell]] of the first non-trivial [[magic square|normal magic square]], called the [[Luoshu Square|''Luoshu'' square]]. All [[integer]]s <math>n \ge 34</math> can be expressed as the sum of five non-zero [[Square number|squares]].<ref>{{Cite book |last1=Niven |first1=Ivan |author1-link=Ivan M. Niven |title=An Introduction to the Theory of Numbers |last2=Zuckerman |first2=Herbert S. |last3=Montgomery |first3=Hugh L. |author3-link=Hugh Lowell Montgomery |publisher=[[Wiley (publisher)|John Wiley]] |year=1980 |isbn=978-0-19-853171-5 |edition=5th |location=New York, NY |pages=144, 145}}</ref><ref>{{Cite OEIS|A047701|All positive numbers that are not the sum of 5 nonzero squares.|access-date=2023-09-20}}

{{Bullet list
|A [[120-cell|regular 120-cell]], the dual ''polychoron'' to the regular [[600-cell]], can fit one hundred and twenty 5-cells. Also, five [[24-cell]]s fit inside a [[small stellated 120-cell]], the first [[stellation]] of the 120-cell. <br />A subset of the vertices of the small stellated 120-cell are matched by the [[great duoantiprism]] star, which is the only [[Uniform polytope|uniform]] nonconvex [[duoantiprism|''duoantiprismatic'']] solution in the fourth dimension, constructed from the [[polytope]] [[cartesian product]] <math>\{5\}\otimes\{5/3\}</math> and made of fifty [[tetrahedra]], ten [[pentagrammic crossed antiprism]]s, ten [[pentagonal antiprism]]s, and fifty vertices.<ref name="RegPoly3" />{{rp|p.124}}

|The [[Abstract polytope|abstract]] four-dimensional [[57-cell]] is made of fifty-seven [[Hemi-icosahedron|hemi-icosahedral]] cells, in-which five surround each edge.<ref>{{Cite journal |last = Coxeter |first = H. S. M. |author-link = Harold Scott MacDonald Coxeter |doi = 10.1007/BF00149428 |issue = 1 |journal = [[Geometriae Dedicata]] |mr = 679218 |pages = 87–99 |title = Ten toroids and fifty-seven hemidodecahedra |volume = 13 |year = 1982 |s2cid = 120672023 }}.</ref> The [[11-cell]], another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven [[Hemi-dodecahedron|hemi-dodecahedral cells]] each with fifteen edges.<ref>{{Cite journal |last=Coxeter |first=H. S. M |author-link = Harold Scott MacDonald Coxeter |url=https://www.sciencedirect.com/science/article/abs/pii/S0304020808728147 |title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra |journal=Annals of Discrete Mathematics |series=North-Holland Mathematics Studies |issue=20 |year=1984 |volume=87 |pages=103–114 |doi=10.1016/S0304-0208(08)72814-7 |isbn=978-0-444-86571-7 }}</ref> The [[n-skeleton|skeleton]] of the hemi-dodecahedron is the [[Petersen graph]].
|}}

Overall, the fourth dimension contains five fundamental [[Uniform 4-polytope#Convex uniform 4-polytopes|Weyl groups]] that form a finite number of [[Uniform 4-polytope#Enumeration|uniform polychora]] based on only twenty-five uniform polyhedra: [[A4 polytope|<math>\mathrm A_{4}</math>]], [[B4 polytope|<math>\mathrm B_{4}</math>]], [[D4 polytope|<math>\mathrm D_{4}</math>]], [[F4 polytope|<math>\mathrm F_{4}</math>]], and [[H4 polytope|<math>\mathrm H_{4}</math>]], accompanied by a fifth or sixth general group of unique [[Uniform 4-polytope#Prismatic uniform 4-polytopes|4-prisms]] of Platonic and Archimedean solids. There are also a total of five [[Coxeter group]]s that generate non-prismatic [[Uniform 5-polytope#Regular and uniform honeycombs|Euclidean honeycombs]] in 4-space, alongside five [[Uniform 5-polytope#Regular and uniform hyperbolic honeycombs|compact hyperbolic Coxeter groups]] that generate five regular [[Uniform 5-polytope#Compact regular tessellations of hyperbolic 4-space|compact hyperbolic honeycombs]] with finite [[Facet (geometry)|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 [[Coxeter-Dynkin diagram#Ranks 4–5|rank&nbsp;5]], with [[Coxeter–Dynkin diagram#Paracompact (Koszul simplex groups)|paracompact hyperbolic solutions]] existing through rank 10. Likewise, analogues of four-dimensional <math>\mathrm{H}_{4}</math> [[H4 polytope|hexadecachoric]] or <math>\mathrm{F}_{4}</math> [[List of F4 polytopes|icositetrachoric]] symmetry do not exist in dimensions <math>n</math> ⩾ <math>5</math>; however, there are [[Uniform 5-polytope#Uniform prismatic forms|prismatic groups]] in the fifth dimension which contains [[Prism (geometry)|prisms]] of regular and uniform [[Uniform 4-polytope|4-polytopes]] that have <math>\mathrm{H}_{4}</math> and <math>\mathrm{F}_{4}</math> symmetry. There are also five regular [[List of regular polytopes and compounds#Regular projective 4-polytopes|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.<ref>{{Cite book |last1= McMullen |first1= Peter |author1-link= Peter McMullen |last2= Schulte |first2= Egon |author2-link= Egon Schulte |url = https://archive.org/details/abstractregularp0000mcmu |url-access= registration |title= Abstract Regular Polytopes |publisher= Cambridge University Press |location= Cambridge |series= Encyclopedia of Mathematics and its Applications |volume= 92 |year= 2002 |pages=162–164 |doi = 10.1017/CBO9780511546686 |isbn= 0-521-81496-0 |mr= 1965665 |s2cid=115688843 }}</ref> Only two regular projective polytopes exist in each higher dimensional space.

Generally, [[Star polyhedron|star polytopes]] that are [[Regular polytope|regular]] only exist in [[dimension]]s <math>2</math> ⩽ <math>n</math> < <math>5</math>, and can be constructed using five [[Stellation#Miller's rules|Miller rules]] for stellating polyhedra or higher-dimensional [[polytope]]s.<ref>{{Cite book |last1=Coxeter |first1=H. S. M. |last2=du Val |first2=P. |last3=Flather |first3=H. T. |last4=Petrie |first4=J.F. |author1-link=H. S. M. Coxeter |author2-link=Patrick du Val |display-authors=2 |title=The Fifty-Nine Icosahedra |publisher=[[Springer-Verlag]] |edition=1st |location=New York |year=1982 |pages=7, 8 |doi=10.1007/978-1-4613-8216-4 |isbn=978-0-387-90770-3 |oclc=8667571 |s2cid=118322641}}</ref>

=== Five-dimensional space ===

The [[5-simplex]] or ''hexateron'' is the [[Five-dimensional space|five-dimensional]] analogue of the 5-cell, or 4-simplex. It has Coxeter group <math>\mathrm{A}_{5}</math> as its symmetry group, of order 720&nbsp;=&nbsp;6[[Factorial|!]], whose group structure is represented by the symmetric group <math>\mathrm{S}_{6}</math>, the only finite symmetric group which has an [[outer automorphism]]. The [[5-cube]], made of ten [[tesseract]]s and the 5-cell as its vertex figure, is also regular and one of thirty-one [[uniform 5-polytope]]s under the Coxeter [[Uniform 5-polytope#The B5 family|<math>\mathrm B_{5}</math> hypercubic]] group. The [[demipenteract]], with one hundred and twenty [[Cell (geometry)|cells]], is the only fifth-dimensional [[Semiregular polytope|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 [[Infinity|infinite]] [[Facet (geometry)|facets]] and [[vertex figure]]s; no other regular paracompact honeycombs exist in higher dimensions.<ref>{{cite web|author=H.S.M. Coxeter|url=https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.361.251|title=Regular Honeycombs in Hyperbolic Space |year=1956|page=168|citeseerx=10.1.1.361.251 }}</ref> There are also exclusively twelve [[Complex polytope#Regular complex 5-apeirotopes and higher|complex aperiotopes]] in [[Complex coordinate space|<math>\mathbb{C}^n</math> complex spaces]] of dimensions <math>n</math>&nbsp;⩾&nbsp;<math>5</math>; alongside [[Complex polytope#Enumeration of regular complex 5-polytopes|complex polytopes]] in <math>\mathbb{C}^5</math> and higher under [[simplex]], [[hypercube|hypercubic]] and [[orthoplex]] groups (with [[Complex polytope#van Oss polygon|van Oss polytopes]]).<ref>{{Cite book |author=H. S. M. Coxeter |author-link=H. S. M. Coxeter |title=Regular Complex Polytopes |publisher=Cambridge University Press |edition=2nd |year=1991 |pages=144–146 |doi=10.2307/3617711 |jstor=3617711 |isbn=978-0-521-39490-1 |s2cid=116900933 |zbl=0732.51002 }}</ref>

==== Veronese surface ====

A [[Veronese surface]] in the [[Projective space|projective plane]] <math>\mathbb{P}^5</math> generalizes a [[Pairing|linear]] condition <math>\nu:\mathbb{P}^2\to \mathbb{P}^5</math> for a point to be contained inside a [[conic]], where [[five points determine a conic]].<ref name="conic" />

== In finite simple groups ==
=== Lie groups ===

There are five complex [[exceptional Lie algebra]]s: [[G2 (mathematics)|<math>\mathfrak{g}_2</math>]], [[F4 (mathematics)|<math>\mathfrak{f}_4</math>]], [[E6 (mathematics)|<math>\mathfrak{e}_6</math>]], [[E7 (mathematics)|<math>\mathfrak{e}_7</math>]], and [[E8 (mathematics)|<math>\mathfrak{e}_8</math>]]. The [[Faithful representation|smallest]] of these, <math>\mathfrak{g}_2</math> of [[Real coordinate space|real]] dimension 28, can be represented in five-dimensional complex space and [[Projective geometry|projected]] as a [[Ball (geometry)|ball]] rolling on top of another ball, whose [[motion]] is described in two-dimensional space.<ref>{{Cite journal|title = G<sub>2</sub> and the rolling ball |last1=Baez |first1=John C. |author1-link=John C. Baez|last2=Huerta |first2=John |journal=Trans. Amer. Math. Soc. |volume=366 |issue=10 |year=2014 |pages=5257–5293 |doi=10.1090/s0002-9947-2014-05977-1 |mr=3240924 |s2cid=50818244 |doi-access=free }}</ref> <math>\mathfrak{e}_8</math> is the largest, and holds the other four Lie algebras as [[subgroup]]s, with a representation over <math>\mathbb {R}</math> in dimension 496. It contains an associated [[E8 lattice|lattice]] that is constructed with one hundred and twenty quaternionic [[Icosian|unit icosians]] that make up the vertices of the [[600-cell]], whose Euclidean [[Quaternion#Conjugation, the norm, and reciprocal|norms]] define a quadratic form on a lattice structure [[Isomorphism|isomorphic]] to the optimal configuration of spheres in eight dimensions.<ref>{{Cite journal |last=Baez |first=John C. |author-link=John C. Baez |title=From the Icosahedron to E<sub>8</sub> |journal=London Math. Soc. Newsletter |volume=476 |pages=18–23 |year=2018 |arxiv=1712.06436 |mr=3792329 |s2cid=119151549 |zbl=1476.51020 }}</ref> This [[Sphere packing problem|sphere packing]] <math>\mathrm {E}_{8}</math> lattice structure in [[Uniform 8-polytope|8-space]] is held by the vertex arrangement of the [[5 21 honeycomb|'''5<sub>21</sub>''' honeycomb]], one of five Euclidean honeycombs that admit [[Thorold Gosset|Gosset's]] original definition of a [[Semiregular polytope|semi-regular honeycomb]], which includes the three-dimensional [[Tetrahedral-octahedral honeycomb|alternated cubic honeycomb]].<ref>{{Cite journal |author=H. S. M. Coxeter |author-link=H. S. M. Coxeter |title=Seven Cubes and Ten 24-Cells |url=https://link.springer.com/content/pdf/10.1007/PL00009338.pdf |journal=[[Discrete & Computational Geometry]] |volume=19 |year=1998 |issue=2 |pages=156–157 |doi=10.1007/PL00009338 |doi-access=free |zbl=0898.52004 |s2cid=206861928 }}</ref><ref>{{Cite journal |author=Thorold Gosset |author-link=Thorold Gosset |title=On the regular and semi-regular figures in space of n dimensions |url=https://www.maths.ed.ac.uk/~v1ranick/papers/gosset.pdf |journal=Messenger of Mathematics |volume=29 |year=1900 |pages=43–48 |jfm=30.0494.02 }}</ref> The smallest simple isomorphism found inside finite simple [[Lie group]]s is <math>\mathrm {A_{5}} \cong A_{1}(4) \cong A_{1}(5)</math>,<ref>{{Cite book |last1=Conway |first1=J. H. |author1-link=John Horton Conway |last2=Curtis |first2=R. T. |last3=Norton |first3=S. P. |author3-link=Simon P. Norton |last4=Parker |first4=R. A. |author4-link=Richard A. Parker |last5=Wilson |first5=R. A. |author5-link=Robert Arnott Wilson |title=[[ATLAS of Finite Groups|ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups]] |publisher=[[Clarendon Press]] |page=xv |year=1985 |location=Oxford |isbn=978-0-19-853199-9 |oclc=12106933 |mr=827219 |s2cid=117473588 |zbl=0568.20001 }}</ref> where here <math>\mathrm {A_{n}}</math> represents [[alternating group]]s and <math>A_{n}(q)</math> [[Group of Lie type#Chevalley groups|classical Chevalley groups]]. In particular, the smallest non-solvable group is the alternating group on five letters, which is also the smallest [[Simple group|simple]] non-[[Abelian group|abelian]] group.

=== Sporadic groups ===
[[File:SporadicGroups.png|thumb|300px|This diagram shows the [[subquotient]] relations of the twenty-six '''sporadic groups'''; the five [[Mathieu group]]s form the simplest class (colored red [[File:EllipseSubqR.svg]]). ]]

==== Mathieu groups ====


:Only twelve integers up to [[33 (number)|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.</ref> There are five countably infinite [[Ramsey class]]es of [[permutation]]s.<ref name="ramsey">{{Cite journal |last1=Böttcher |first1=Julia |author1-link=Julia Böttcher |last2=Foniok |first2=Jan |year=2013 |title=Ramsey Properties of Permutations |url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p2 |journal=The Electronic Journal of Combinatorics |volume=20 |issue=1 |page=P2 |arxiv=1103.5686v2 |doi=10.37236/2978 |s2cid=17184541 |zbl=1267.05284}}</ref>{{rp|p.4}}
The five [[Mathieu groups]] constitute the [[Sporadic group#First generation (5 groups): the Mathieu groups|first generation]] in the [[Sporadic groups#Happy Family|happy family]] of [[sporadic groups]]. These are also the first five sporadic groups [[Classification of finite simple groups#Timeline of the proof|to have been described]], defined as <math>\mathrm{M}_{n}</math> [[Mathieu groups#Multiply transitive groups|multiply transitive]] permutation groups on <math>n</math> [[Group object|objects]], with <math>n</math> [[Element (mathematics)|∈]] {11, 12, 22, 23, 24}.<ref name=griess>{{Cite book |author=Robert L. Griess, Jr. |author-link=Robert Griess |title=Twelve Sporadic Groups |series=Springer Monographs in Mathematics |publisher=Springer-Verlag |location=Berlin |year=1998 |pages=1−169 |isbn=978-3-540-62778-4 |doi=10.1007/978-3-662-03516-0 |mr=1707296 |s2cid=116914446 |zbl=0908.20007 }}</ref>{{rp|p.54}} In particular, <math>\mathrm{M}_{11}</math>, the smallest of all sporadic groups, has a [[rank 3 action]] on fifty-five points from an [[Induced representation|induced action]] on [[unordered pair]]s, as well as two [[five-dimensional space|five-dimensional]] [[Faithful representation|faithful complex irreducible representations]] over the [[Field (mathematics)|field]] with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with <math>n</math> elements.<ref>{{Cite journal |last=Jansen |first=Christoph |date=2005 |title=The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups |journal=[[LMS Journal of Computation and Mathematics]] |volume=8 |pages=123–124 |publisher=[[London Mathematical Society]] |doi=10.1112/S1461157000000930 |doi-access=free |mr=2153793 |s2cid=121362819 |zbl=1089.20006 }}</ref> Of precisely five different [[conjugacy class]]es of [[maximal subgroup]]s of <math>\mathrm{M}_{11}</math>, one is the [[Almost simple group|almost simple]] symmetric group [[Symmetric group#Low degree groups|<math>\mathrm{S}_5</math>]] (of order 5[[Factorial|!]]), and another is <math>\mathrm{M}_{10}</math>, also almost simple, that functions as a [[point stabilizer]] which contains five as its largest [[prime factor]] in its [[group order]]: {{math|1=2<sup>4</sup>·3<sup>2</sup>·5 = [[2]]·[[3]]·[[4]]·5·[[6]] = [[8]]·[[9]]·[[10]] = 720}}. On the other hand, whereas <math>\mathrm{M}_{11}</math> is sharply 4-transitive, <math>\mathrm{M}_{12}</math> is [[Mathieu groups#Multiply transitive groups|sharply 5-transitive]] and <math>\mathrm{M}_{24}</math> is 5-transitive, and as such they are the only two 5-transitive groups that are not [[symmetric group]]s or [[alternating group]]s.<ref>{{Cite book |last=Cameron |first=Peter J. |title=Projective and Polar Spaces |chapter=Chapter 9: The geometry of the Mathieu groups |chapter-url=https://webspace.maths.qmul.ac.uk/p.j.cameron/pps/pps9.pdf |publisher=University of London, Queen Mary and Westfield College |year=1992 |page=139|isbn=978-0-902-48012-4 |s2cid=115302359 }}</ref> <math>\mathrm{M}_{22}</math> has the first five prime numbers as its distinct prime factors in its order of {{math|1=2<sup>7</sup>·3<sup>2</sup>·5·[[7]]·[[11 (number)|11]]}}; all Mathieu groups are subgroups of <math>\mathrm{M}_{24}</math>, which under the [[Witt design]] <math>\mathrm{W}_{24}</math> of [[Steiner system#The Steiner system S(5, 8, 24)|Steiner system]] <math>\operatorname{S(5, 8, 24)}</math> emerges a construction of the [[Binary Golay code|extended binary Golay code]] <math>\mathrm{B}_{24}</math> that has <math>\mathrm{M}_{24}</math> as its [[automorphism group]].<ref name=griess />{{rp|pp.39,47,55}} <math>\mathrm{W}_{24}</math> generates ''octads'' from [[Code word (communication)|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.<ref name=griess />{{rp|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]] '''Λ<sub>24</sub>''', which is primarily constructed using the [[Weyl character formula#Statement of Weyl character formula|Weyl vector]] <math>(0,1,2,3, \dots ,24; 70)</math> that admits the only non-unitary solution to the ''[[cannonball problem]]'', where the sum of the [[Square number|squares]] of the first twenty-four integers is equivalent to the square of another integer, the fifth [[pentatope number]] (70). The [[subquotient]]s of the automorphism of the Leech lattice, [[Conway group]] <math>\mathrm{Co}_{0}</math>, is in turn the subject of the [[Sporadic group#Second generation (7 groups): the Leech lattice|second generation]] of seven sporadic groups.<ref name=griess />{{rp|pp.99,125}}


5 is [[conjecture]]d to be the only [[Parity (mathematics)|odd]], [[untouchable number]]; if this is the case, then five will be the only odd prime number that is not the base of an [[Aliquot sequence|aliquot]] tree.<ref>{{Cite journal |last1=Pomerance |first1=Carl |last2=Yang |first2=Hee-Sung |date=14 June 2012 |title=On Untouchable Numbers and Related Problems |url=https://math.dartmouth.edu/~carlp/uupaper3.pdf |publisher=[[Dartmouth College]] |page=1 |s2cid=30344483 |website=math.dartmouth.edu}} 2010 Mathematics Subject Classification. 11A25, 11Y70, 11Y16.</ref>[[File:SporadicGroups.png|thumb|300px|This diagram shows the [[subquotient]] relations of the twenty-six '''sporadic groups'''; the five [[Mathieu group]]s form the simplest class (colored red [[File:EllipseSubqR.svg]]). ]]Every odd number greater than five is conjectured to be expressible as the sum of three prime numbers; [[Harald Andres Helfgott|Helfgott]] has provided a proof of this<ref>{{Cite book |last=Helfgott |first=Harald Andres |title=Seoul [[International Congress of Mathematicians]] Proceedings |date=2014 |publisher=Kyung Moon SA |isbn=978-89-6105-805-6 |editor-last=Jang |editor-first=Sun Young |volume=2 |location=Seoul, KOR |pages=391–418 |chapter=The ternary Goldbach problem |oclc=913564239 |chapter-url=https://www.imj-prg.fr/wp-content/uploads/2020/prix/helfgott2014.pdf}}</ref> (also known as the [[Goldbach's weak conjecture|odd Goldbach conjecture]]) that is already widely acknowledged by mathematicians as it still undergoes [[peer-review]]. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).<ref>{{Cite journal |last=Tao |first=Terence |date=March 2014 |title=Every odd number greater than 1 has a representation is the sum of at most five primes |url=https://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02733-0/S0025-5718-2013-02733-0.pdf |journal=Mathematics of Computation |volume=83 |pages=997–1038 |doi=10.1090/S0025-5718-2013-02733-0 |mr=3143702 |s2cid=2618958 |number=286}}</ref>{{Unsolved|mathematics|Is 5 the only odd, untouchable number?}}
==== Harada-Norton group ====
=== Group Theory ===
In [[graph theory]], all [[Graph theory|graphs]] with four or fewer vertices are [[Planar graph|planar]], however, there is a graph with five vertices that is not: ''K''<sub>5</sub>, the [[complete graph]] with five vertices. By [[Kuratowski's theorem]], a finite graph is planar [[iff|if and only if]] it does not contain a subgraph that is a subdivision of , or ''K''<sub>3,3</sub>, the [[utility graph]].<ref>{{Cite journal |last=Burnstein |first=Michael|title=Kuratowski-Pontrjagin theorem on planar graphs |journal=[[Journal of Combinatorial Theory]] | series=Series B |volume=24 |issue=2 |year=1978 |pages=228–232 |doi= 10.1016/0095-8956(78)90024-2 |doi-access=free }}</ref>


A [[Centralizer and normalizer|centralizer]] of an element of order 5 inside the [[Monster group|largest sporadic group]] <math>\mathrm {F_1}</math> arises from the product between [[Harada–Norton group|Harada–Norton]] sporadic group <math>\mathrm{HN}</math> and a group of order 5.<ref>{{Cite journal |last1=Lux |first1=Klaus |last2=Noeske |first2=Felix |last3=Ryba |first3=Alexander J. E. |title=The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2 |journal=[[Journal of Algebra]] |publisher=[[Elsevier]] |volume=319 |issue=1 |year=2008 |location=Amsterdam |pages=320–335 |doi=10.1016/j.jalgebra.2007.03.046 |doi-access=free |mr=2378074 |s2cid=120706746 |zbl=1135.20007 }}</ref><ref>{{Cite journal |last=Wilson |first=Robert A. |author-link=Robert Arnott Wilson |title=The odd local subgroups of the Monster |journal=Journal of Australian Mathematical Society (Series A) |publisher=[[Cambridge University Press]] |volume=44 |issue=1 |pages=12–13 |year=2009 |location=Cambridge |doi=10.1017/S1446788700031323 |doi-access=free |mr=914399 |s2cid=123184319 |zbl=0636.20014 }}</ref> On its own, <math>\mathrm{HN}</math> can be represented using [[Generator (mathematics)|standard generators]] <math>(a,b,ab)</math> that further dictate a condition where <math>o([a, b]) = 5</math>.<ref>{{Cite book |last=Wilson |first=R.A |author-link=Robert Arnott Wilson |title=The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249) |chapter=An Atlas of Sporadic Group Representations |chapter-url=https://webspace.maths.qmul.ac.uk/r.a.wilson/pubs_files/ASGRweb.pdf |publisher=Cambridge University Press |location=Cambridge |year=1998 |page=267 |doi=10.1017/CBO9780511565830.024 |isbn=978-0-511-56583-0 |oclc=726827806 |zbl=0914.20016 |s2cid=59394831 }}</ref><ref>{{Cite journal |last1=Nickerson |first1=S.J. |last2=Wilson |first2=R.A. |author2-link=Robert Anton Wilson |title=Semi-Presentations for the Sporadic Simple Groups |journal=Experimental Mathematics |volume=14 |issue=3 |page=367 |publisher=[[Taylor & Francis]] |year=2011 |location=Oxfordshire |doi=10.1080/10586458.2005.10128927 |mr=2172713 |zbl=1087.20025 |s2cid=13100616 }}</ref> This condition is also held by other generators that belong to the [[Tits group]] <math>\mathrm{T}</math>,<ref>{{Cite web |last1=Wilson |first1=R.A. |author1-link=Robert Arnott Wilson |last2=Parker |first2=R.A. |author2-link=Richard A. Parker |last3=Nickerson |first3=S.J. |last4=Bray |first4=J.N. |title=Exceptional group <sup>2</sup>F<sub>4</sub>(2)', Tits group T |website=ATLAS of Finite Group Representations |url=https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TF42/ |year=1999}}</ref> the only [[finite simple group]] that is a ''non-strict'' group of Lie type that can also classify as sporadic. Furthermore, over the field with five elements, <math>\mathrm{HN}</math> holds a 133-dimensional representation where 5 acts on a [[Commutative property|commutative]] yet non-[[Associative property|associative]] product as a 5-[[Modular arithmetic|modular]] analogue of the [[Griess algebra]] <math>V_{2}</math><sup>{{large|♮}}</sup>,<ref>{{Cite journal |last1=Ryba |first1=A. J. E. |title=A natural invariant algebra for the Harada-Norton group |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |publisher=[[Cambridge University Press]] |volume=119 |issue=4 |location=Cambridge |year=1996 |pages=597–614 |doi=10.1017/S0305004100074454 |bibcode=1996MPCPS.119..597R |mr=1362942 |s2cid=119931824 |zbl=0851.20034 }}</ref> which holds <math>\mathrm {F_1}</math> as its [[automorphism group]].
There are five complex [[exceptional Lie algebra]]s. The five [[Mathieu groups]] constitute the [[Sporadic group#First generation (5 groups): the Mathieu groups|first generation]] in the [[Sporadic groups#Happy Family|happy family]] of [[sporadic groups]]. These are also the first five sporadic groups [[Classification of finite simple groups#Timeline of the proof|to have been described]].<ref name="griess">{{Cite book |author=Robert L. Griess, Jr. |author-link=Robert Griess |title=Twelve Sporadic Groups |publisher=Springer-Verlag |year=1998 |isbn=978-3-540-62778-4 |series=Springer Monographs in Mathematics |location=Berlin |pages=1−169 |doi=10.1007/978-3-662-03516-0 |mr=1707296 |zbl=0908.20007 |s2cid=116914446}}</ref>{{rp|p.54}} A [[Centralizer and normalizer|centralizer]] of an element of order 5 inside the [[Monster group|largest sporadic group]] <math>\mathrm {F_1}</math> arises from the product between [[Harada–Norton group|Harada–Norton]] sporadic group <math>\mathrm{HN}</math> and a group of order 5.<ref>{{Cite journal |last1=Lux |first1=Klaus |last2=Noeske |first2=Felix |last3=Ryba |first3=Alexander J. E. |year=2008 |title=The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2 |journal=[[Journal of Algebra]] |location=Amsterdam |publisher=[[Elsevier]] |volume=319 |issue=1 |pages=320–335 |doi=10.1016/j.jalgebra.2007.03.046 |mr=2378074 |s2cid=120706746 |zbl=1135.20007 |doi-access=free}}</ref><ref>{{Cite journal |last=Wilson |first=Robert A. |author-link=Robert Arnott Wilson |year=2009 |title=The odd local subgroups of the Monster |journal=Journal of Australian Mathematical Society (Series A) |location=Cambridge |publisher=[[Cambridge University Press]] |volume=44 |issue=1 |pages=12–13 |doi=10.1017/S1446788700031323 |mr=914399 |s2cid=123184319 |zbl=0636.20014 |doi-access=free}}</ref>


== List of basic calculations ==
== List of basic calculations ==
Line 147: Line 95:
!9
!9
!10
!10
! style="width:5px;"|
!11
!11
!12
!12
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|[[45 (number)|45]]
|[[45 (number)|45]]
|[[50 (number)|50]]
|[[50 (number)|50]]
! style="width:5px;"|
|[[55 (number)|55]]
|[[55 (number)|55]]
|[[60 (number)|60]]
|[[60 (number)|60]]
Line 196: Line 142:
!9
!9
!10
!10
! style="width:5px;"|
!11
!11
!12
!12
Line 214: Line 159:
|0.{{overline|5}}
|0.{{overline|5}}
|0.5
|0.5
!
|0.{{overline|45}}
|0.{{overline|45}}
|0.41{{overline|6}}
|0.41{{overline|6}}
Line 231: Line 175:
|1.8
|1.8
|[[2]]
|[[2]]
!
|2.2
|2.2
|2.4
|2.4
Line 252: Line 195:
!9
!9
!10
!10
! style="width:5px;"|
!11
!11
!12
!12
Line 270: Line 212:
|1953125
|1953125
|9765625
|9765625
!
|48828125
|48828125
|244140625
|244140625
Line 287: Line 228:
|59049
|59049
|100000
|100000
!
|161051
|161051
|248832
|248832
Line 294: Line 234:
|759375
|759375
|}
|}

=== Decimal properties ===

All multiples of 5 will end in either 5 or {{num|0}}, and [[Fraction (mathematics)#Vulgar, proper, and improper fractions|vulgar fractions]] with 5 or {{num|2}} in the [[fraction (mathematics)|denominator]] do not yield infinite [[decimal]] expansions because they are prime factors of [[10]], the base.

In the [[Power (mathematics)|powers]] of 5, every power ends with the number five, and from 5<sup>3</sup> onward, if the exponent is [[Parity (mathematics)|odd]], then the hundreds digit is [[1]], and if it is even, the hundreds digit is [[6]].

A number <math>n</math> raised to the fifth power always ends in the same digit as <math>n</math>.


== Evolution of the Arabic digit ==
== Evolution of the Arabic digit ==
Line 344: Line 276:


==== Islam ====
==== Islam ====
The [[Five Pillars of Islam]].<ref>{{Cite web|title=PBS – Islam: Empire of Faith – Faith – Five Pillars|url=https://www.pbs.org/empires/islam/faithpillars.html|access-date=2020-08-03|website=www.pbs.org}}</ref>
The [[Five Pillars of Islam]].<ref>{{Cite web|title=PBS – Islam: Empire of Faith – Faith – Five Pillars|url=https://www.pbs.org/empires/islam/faithpillars.html|access-date=2020-08-03|website=www.pbs.org}}</ref> The [[Five-pointed star|five-pointed]] [[Simple polygon|simple]] star ☆ is one of the five used in Islamic [[Girih tiles]].<ref>{{Cite journal |last=Sarhangi |first=Reza |year=2012 |title=Interlocking Star Polygons in Persian Architecture: The Special Case of the Decagram in Mosaic Designs |url=https://link.springer.com/content/pdf/10.1007/s00004-012-0117-5.pdf |journal=Nexus Network Journal |volume=14 |issue=2 |page=350 |doi=10.1007/s00004-012-0117-5 |s2cid=124558613 |doi-access=free}}</ref>


=== Mysticism ===
=== Mysticism ===

==== Gnosticism ====
==== Gnosticism ====
The number five was an important symbolic number in [[Manichaeism]], with heavenly beings, concepts, and others often grouped in sets of five.
The number five was an important symbolic number in [[Manichaeism]], with heavenly beings, concepts, and others often grouped in sets of five.{{Citation needed|date=November 2024}}


==== Alchemy ====
==== Alchemy ====
Line 365: Line 296:
== See also ==
== See also ==
{{Portal|Mathematics}}
{{Portal|Mathematics}}

[[5 (disambiguation)]]
* [[5 (disambiguation)]]
{{clear}}
{{clear}}

== Notes ==
<references />

== References ==
== References ==
{{Reflist}}
{{Reflist}}

Latest revision as of 16:44, 12 November 2024

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).
← 4 5 6 →
−1 0 1 2 3 4 5 6 7 8 9 →
Cardinalfive
Ordinal5th (fifth)
Numeral systemquinary
Factorizationprime
Prime3rd
Divisors1, 5
Greek numeralΕ´
Roman numeralV, v
Greek prefixpenta-/pent-
Latin prefixquinque-/quinqu-/quint-
Binary1012
Ternary123
Senary56
Octal58
Duodecimal512
Hexadecimal516
Greekε (or Ε)
Arabic, Kurdish٥
Persian, Sindhi, Urdu۵
Ge'ez፭
Bengali৫
Kannada೫
Punjabi੫
Chinese numeral五
ArmenianԵ
Devanāgarī५
Hebrewה
Khmer៥
Telugu౫
Malayalam൫
Tamil௫
Thai๕
Babylonian numeral𒐙
Egyptian hieroglyph, Chinese counting rod|||||
Maya numerals𝋥
Morse code.....

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.

Humans, and many other animals, have 5 digits on their limbs.

Mathematics

[edit]
The first Pythagorean triple

5 is a Fermat prime, a Mersenne prime exponent, as well as a Fibonacci number. 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5).[1]

5 is the first safe prime,[2] and the first good prime.[3] 11 forms the first pair of sexy primes with 5.[4] 5 is the second Fermat prime, of a total of five known Fermat primes.[5] 5 is also the first of three known Wilson primes (5, 13, 563).[6]

Geometry

[edit]

A shape with five sides is called a pentagon. The pentagon is the first regular polygon that does not tile the plane with copies of itself. It is the largest face any of the five regular three-dimensional regular Platonic solid can have.

A conic is determined using five points in the same way that two points are needed to determine a line.[7] A pentagram, or five-pointed polygram, is a star polygon constructed by connecting some non-adjacent of a regular pentagon as self-intersecting edges.[8] The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol {5/2}) appears prominently in Penrose tilings. Pentagrams are facets inside Kepler–Poinsot star polyhedra and Schläfli–Hess star polychora.

There are five regular Platonic solids the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.[9]

The chromatic number of the plane is 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.[10] Five is a lower depending for the chromatic number of the plane, but this may depend on the choice of set-theoretical axioms:[11]

The plane contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations. Uniform tilings of the plane, are generated from combinations of only five regular polygons.[12]

Higher Dimensional Geometry

[edit]

A hypertetrahedron, or 5-cell, is the 4 dimensional analogue of the tetrahedron. It has five vertices. Its orthographic projection is homomorphic to the group K5.[13]: p.120 

There are five fundamental mirror symmetry point group families in 4-dimensions. There are also 5 compact hyperbolic Coxeter groups, or 4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.[14]

The four-dimensional 5-cell is the simplest regular polychoron.

Algebra

[edit]
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. All integers can be expressed as the sum of five non-zero squares.[15][16] There are five countably infinite Ramsey classes of permutations.[17]: p.4  5 is conjectured to be the only odd, untouchable number; if this is the case, then five will be the only odd prime number that is not the base of an aliquot tree.[18]

This diagram shows the subquotient relations of the twenty-six sporadic groups; the five Mathieu groups form the simplest class (colored red ).

Every odd number greater than five is conjectured to be expressible as the sum of three prime numbers; Helfgott has provided a proof of this[19] (also known as the odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes peer-review. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).[20]

Unsolved problem in mathematics:
Is 5 the only odd, untouchable number?
(more unsolved problems in mathematics)

Group Theory

[edit]

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. By Kuratowski's theorem, a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of , or K3,3, the utility graph.[21]

There are five complex exceptional Lie algebras. 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.[22]: p.54  A centralizer of an element of order 5 inside the largest sporadic group arises from the product between Harada–Norton sporadic group and a group of order 5.[23][24]

List of basic calculations

[edit]
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
5x 5 25 125 625 3125 15625 78125 390625 1953125 9765625 48828125 244140625 1220703125 6103515625 30517578125
x5 1 32 243 1024 7776 16807 32768 59049 100000 161051 248832 371293 537824 759375

Evolution of the Arabic digit

[edit]

The evolution of the modern Western digit for the numeral for five is traced back to the Indian system of numerals, where on some earlier versions, the numeral bore resemblance to variations of the number four, rather than "5" (as it is represented today). The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. Later on, Arabic traditions transformed the digit in several ways, producing forms that were still similar to the numeral for four, with similarities to the numeral for three; yet, still unlike the modern five.[25] It was from those digits that Europeans finally came up with the modern 5 (represented in writings by Dürer, for example).

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.

Other fields

[edit]

Astronomy

[edit]

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

Biology

[edit]

There are usually considered to be five senses (in general terms); the five basic tastes are sweet, salty, sour, bitter, and umami.[26] Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.[27] Five is the number of appendages on most starfish, which exhibit pentamerism.[28]

Computing

[edit]

5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.[29]

Literature

[edit]

Poetry

[edit]

A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.[30]

Music

[edit]

Modern musical notation uses a musical staff made of five horizontal lines.[31] A scale with five notes per octave is called a pentatonic scale.[32] A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[33] 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

[edit]

Judaism

[edit]

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").[34] 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.[35]

Christianity

[edit]

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).[36]

Islam

[edit]

The Five Pillars of Islam.[37] The five-pointed simple star ☆ is one of the five used in Islamic Girih tiles.[38]

Mysticism

[edit]

Gnosticism

[edit]

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

Alchemy

[edit]

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). In East Asian tradition, there are five elements: water, fire, earth, wood, and metal.[39] 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.[40] 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. There are also five elements in the traditional Chinese Wuxing.[41]

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.[42] The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.

Miscellaneous fields

[edit]
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).[43]
  • The number of dots in a quincunx.[44]

See also

[edit]

Notes

[edit]
  1. ^ Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  2. ^ 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.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  4. ^ 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.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A019434 (Fermat primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21.
  6. ^ 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.
  7. ^ Dixon, A. C. (March 1908). "The Conic through Five Given Points". The Mathematical Gazette. 4 (70). The Mathematical Association: 228–230. doi:10.2307/3605147. JSTOR 3605147. S2CID 125356690.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A307681 (Difference between the number of sides and the number of diagonals of a convex n-gon.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61
  10. ^ 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.
  11. ^ Exoo, Geoffrey; Ismailescu, Dan (2020). "The Chromatic Number of the Plane is At Least 5: A New Proof". Discrete & Computational Geometry. 64. New York, NY: Springer: 216–226. arXiv:1805.00157. doi:10.1007/s00454-019-00058-1. MR 4110534. S2CID 119266055. Zbl 1445.05040.
  12. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. 50 (5). Taylor & Francis, Ltd.: 227–236. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
  13. ^ H. S. M. Coxeter (1973). Regular Polytopes (3rd ed.). New York: Dover Publications, Inc. pp. 1–368. ISBN 978-0-486-61480-9.
  14. ^ 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.
  15. ^ 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.
  16. ^ 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.
  17. ^ 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.
  18. ^ Pomerance, Carl; Yang, Hee-Sung (14 June 2012). "On Untouchable Numbers and Related Problems" (PDF). math.dartmouth.edu. Dartmouth College: 1. S2CID 30344483. 2010 Mathematics Subject Classification. 11A25, 11Y70, 11Y16.
  19. ^ 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.
  20. ^ Tao, Terence (March 2014). "Every odd number greater than 1 has a representation 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.
  21. ^ 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.
  22. ^ 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.
  23. ^ 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. 319 (1). Amsterdam: Elsevier: 320–335. doi:10.1016/j.jalgebra.2007.03.046. MR 2378074. S2CID 120706746. Zbl 1135.20007.
  24. ^ Wilson, Robert A. (2009). "The odd local subgroups of the Monster". Journal of Australian Mathematical Society (Series A). 44 (1). Cambridge: Cambridge University Press: 12–13. doi:10.1017/S1446788700031323. MR 0914399. S2CID 123184319. Zbl 0636.20014.
  25. ^ 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
  26. ^ 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...
  27. ^ 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.
  28. ^ 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
  29. ^ Pozrikidis, Constantine (2012-09-17). XML in Scientific Computing. CRC Press. p. 209. ISBN 978-1-4665-1228-3. 5 5 005 ENQ (enquiry)
  30. ^ 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)
  31. ^ "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
  32. ^ 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
  33. ^ 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
  34. ^ Pelaia, Ariela. "Judaism 101: What Are the Five Books of Moses?". Learn Religions. Retrieved 2020-08-03.
  35. ^ Zenner, Walter P. (1988-01-01). Persistence and Flexibility: Anthropological Perspectives on the American Jewish Experience. SUNY Press. p. 284. ISBN 978-0-88706-748-8.
  36. ^ "CATHOLIC ENCYCLOPEDIA: The Five Sacred Wounds". www.newadvent.org. Retrieved 2020-08-02.
  37. ^ "PBS – Islam: Empire of Faith – Faith – Five Pillars". www.pbs.org. Retrieved 2020-08-03.
  38. ^ 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.
  39. ^ Yoon, Hong-key (2006). The Culture of Fengshui in Korea: An Exploration of East Asian Geomancy. Lexington Books. p. 59. ISBN 978-0-7391-1348-6. The first category is the Five Agents [Elements] namely, Water, Fire, Wood, Metal, and Earth.
  40. ^ Walsh, Len (2008-11-15). Read Japanese Today: The Easy Way to Learn 400 Practical Kanji. Tuttle Publishing. ISBN 978-1-4629-1592-7. The Japanese names of the days of the week are taken from the names of the seven basic nature symbols
  41. ^ Chen, Yuan (2014). "Legitimation Discourse and the Theory of the Five Elements in Imperial China". Journal of Song-Yuan Studies. 44 (1): 325–364. doi:10.1353/sys.2014.0000. ISSN 2154-6665. S2CID 147099574.
  42. ^ Kronland-Martinet, Richard; Ystad, Sølvi; Jensen, Kristoffer (2008-07-19). Computer Music Modeling and Retrieval. Sense of Sounds: 4th International Symposium, CMMR 2007, Copenhagen, Denmark, August 2007, Revised Papers. Springer. p. 502. ISBN 978-3-540-85035-9. Plato and Aristotle postulated a fifth state of matter, which they called "idea" or quintessence" (from "quint" which means "fifth")
  43. ^ "Olympic Rings – Symbol of the Olympic Movement". International Olympic Committee. 2020-06-23. Retrieved 2020-08-02.
  44. ^ Laplante, Philip A. (2018-10-03). Comprehensive Dictionary of Electrical Engineering. CRC Press. p. 562. ISBN 978-1-4200-3780-7. quincunx five points

References

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Further reading

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