5

File:Pythagoras' Special Triples.svg ]]

5 is a Fermat prime, a Mersenne prime exponent, {{Cite OEIS|A000043|mersenne prime exponents}} 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).{{Cite OEIS|1=A003273 |2=Congruent numbers|access-date=2016-06-01}}

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

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.{{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 }} A pentagram, or five-pointed polygram, is a star polygon constructed by connecting some non-adjacent of a regular pentagon as self-intersecting edges.{{Cite OEIS|A307681|Difference between the number of sides and the number of diagonals of a convex n-gon.}} The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol {{math|1={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.Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61

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.{{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 |archive-date=2016-03-03 |access-date=2023-01-18 |archive-url=https://web.archive.org/web/20160303235526/http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |url-status=dead }}

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 K_5.{{Cite book |author=H. S. M. Coxeter |author-link=Harold Scott MacDonald Coxeter |title=Regular Polytopes |publisher=Dover Publications, Inc. |year=1973 |isbn=978-0-486-61480-9 |edition=3rd |location=New York |pages=1–368}}{{rp|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.{{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}}

File:Schlegel wireframe 5-cell.png is the simplest regular polychoron.]]

File:Magic Square Lo Shu.svg]]5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. All integers $n\; \backslash ge\; 34$ can be expressed as the sum of five non-zero squares.{{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=John Wiley |year=1980 |isbn=978-0-19-853171-5 |edition=5th |location=New York, NY |pages=144, 145}}{{Cite OEIS|A047701|All positive numbers that are not the sum of 5 nonzero squares.|access-date=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. There are five countably infinite Ramsey classes of permutations.{{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}}{{rp|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.{{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.File:SporadicGroups.png relations of the twenty-six sporadic groups; the five Mathieu groups 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; Helfgott has provided a proof of this{{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}} (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).{{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}}{{Unsolved|mathematics|Is 5 the only odd, untouchable number?}}

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 K₅, or K_3,3, the utility graph.{{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 }}

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.{{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}}{{rp|p.54}} A centralizer of an element of order 5 inside the largest sporadic group $\backslash mathrm\; \{F\_1\}$ arises from the product between Harada–Norton sporadic group $\backslash mathrm\{HN\}$ and a group of order 5.{{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}}{{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}}

File:Seven-segment 5.svgThe 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.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 It was from those digits that Europeans finally came up with the modern 5 (represented in writings by Dürer, for example).

File:Evolution5glyph.png

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 45px.

On the seven-segment display of a calculator and digital clock, it is often 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. This makes it often indistinguishable from the letter S. Higher segment displays may sometimes may make use of a diagonal for one of the two.

Judaïsm

Five is according to Maharal of Prague the number defined as the center point which unifies four extremes.

The Five Pillars of Islam.{{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}} The five-pointed simple star ☆ is one of the five used in Islamic Girih tiles.{{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}}

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• 5 (disambiguation)

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• {{Cite book |last=Wells |first=D. |title=The Penguin Dictionary of Curious and Interesting Numbers |title-link=The Penguin Dictionary of Curious and Interesting Numbers |location=London, UK |publisher=Penguin Group |date=1987 |pages=58–67}}

• [https://primes.utm.edu/curios/page.php/5.html Prime curiosities: 5]
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Category:Integers

Category:5 (number)