5

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title: "5" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["integers", "5-(number)"] description: "Natural number" topic_path: "general/integers" source: "https://en.wikipedia.org/wiki/5" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Natural number ::

::data[format=table title="infobox number"]

Field	Value
number	5
numeral	quinary
prime	3rd
divisor	1, 5
roman	V, v
greek prefix	penta-/pent-
latin prefix	quinque-/quinqu-/quint-
lang1	Greek
lang1 symbol	ε (or Ε)
lang2	Arabic, Kurdish
lang2 symbol
lang3	Persian, Sindhi, Urdu
lang3 symbol
lang4	Ge'ez
lang4 symbol	፭
lang5	Bengali
lang5 symbol
lang6	Kannada
lang6 symbol
lang7	Punjabi
lang7 symbol
lang8	Chinese numeral
lang8 symbol	五
lang9	Armenian
lang10 symbol
lang11	Hebrew
lang11 symbol
lang12	Khmer
lang12 symbol	៥
lang13	Telugu
lang13 symbol
lang14	Malayalam
lang14 symbol
lang15	Tamil
lang15 symbol
lang16	Thai
lang16 symbol	๕
lang21	ASCII value
lang21 symbol	ENQ
lang22	Santali
::

::callout[type=note] This article is about the number. For other uses, see 5 (disambiguation), Number Five (disambiguation), and The Five (disambiguation). ::

|number=5 |numeral=quinary |prime=3rd |divisor=1, 5 |roman =V, v |greek prefix=penta-/pent- |latin prefix=quinque-/quinqu-/quint- |lang1=Greek |lang1 symbol=ε (or Ε) |lang2=Arabic, Kurdish |lang2 symbol= |lang3=Persian, Sindhi, Urdu |lang3 symbol= |lang4=Ge'ez |lang4 symbol=፭ |lang5=Bengali |lang5 symbol= |lang6=Kannada |lang6 symbol= |lang7=Punjabi |lang7 symbol= |lang8=Chinese numeral |lang8 symbol=五 |lang9=Armenian|lang9 symbol=Ե|lang10=Devanāgarī |lang10 symbol= |lang11=Hebrew |lang11 symbol= |lang12=Khmer |lang12 symbol=៥ |lang13=Telugu |lang13 symbol= |lang14=Malayalam |lang14 symbol= |lang15=Tamil |lang15 symbol= |lang16=Thai |lang16 symbol=๕|lang17=Babylonian numeral|lang17 symbol=𒐙|lang18=Egyptian hieroglyph, Chinese counting rod|lang18 symbol=|lang19=Maya numerals|lang19 symbol=𝋥|lang20=Morse code|lang20 symbol= |lang21=ASCII value |lang21 symbol=ENQ |lang22=Santali|lang22 symbol=|cardinal=five|ordinal=5th (fifth)}}

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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/69/Pythagoras'_Special_Triples.svg" caption="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).

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

Geometry

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

A conic is determined using five points in the same way that two points are needed to determine a line. A pentagram, or five-pointed polygram, is a star polygon constructed by connecting some non-adjacent vertices of a regular pentagon as self-intersecting edges. The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol ) 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.

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.

Higher dimensional geometry

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.

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.

::figure[src="https://upload.wikimedia.org/wikipedia/commons/5/5a/Schlegel_wireframe_5-cell.png" caption="The four-dimensional [[5-cell]] is the simplest regular [[polychoron]]."] ::

Algebra

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/af/Magic_Square_Lo_Shu.svg" caption="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.

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.[[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; Helfgott has provided a proof of this (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).

Group theory

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 K3,3, the utility graph.

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. A centralizer of an element of order 5 inside the largest sporadic group \mathrm {F_1} arises from the product between Harada–Norton sporadic group \mathrm{HN} and a group of order 5.

List of basic calculations

::data[format=table]

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
::

::data[format=table]

Division	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
'*5 ÷ *x'''''	5	2.5	1.	1.25	1	0.8	0.	0.625	0.	0.5	0.	0.41	0.	0.3	0.
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
::

::data[format=table]

Exponentiation	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
5	5	25	125	625	3125	15625	78125	390625	1953125	9765625	48828125	244140625	1220703125	6103515625	30517578125
x	1	32	243	1024	7776	16807	32768	59049	100000	161051	248832	371293	537824	759375
::

Evolution of the Arabic digit

::figure[src="https://upload.wikimedia.org/wikipedia/commons/5/50/Seven-segment_5.svg" caption="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)."] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/69/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 [[File:Text figures 256.svg|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.

Other fields

In Basque, bost, "5", also means "a lot".

Religion

Judaism

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

Islam

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

References

References

1. {{Cite OEIS. A000043. mersenne prime exponents
2. {{Cite OEIS
3. {{Cite OEIS. A005385. Safe primes p: (p-1)/2 is also prime
4. {{Cite OEIS. A028388. Good primes
5. {{Cite OEIS. A023201. Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)
6. {{Cite OEIS. A019434. Fermat primes
7. {{Cite OEIS. A007540. Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).
8. Dixon, A. C.. (March 1908). "The Conic through Five Given Points". The Mathematical Association.
9. {{Cite OEIS. A307681. Difference between the number of sides and the number of diagonals of a convex n-gon.
10. Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 61
11. (November 1977). "Tilings by Regular Polygons". Taylor & Francis, Ltd..
12. H. S. M. Coxeter. (1973). "[[Regular Polytopes (book)". [[Dover Publications, Inc.]].
13. (2002). "Abstract Regular Polytopes". Cambridge University Press.
14. (1980). "An Introduction to the Theory of Numbers". [[Wiley (publisher).
15. (2013). "Ramsey Properties of Permutations". The Electronic Journal of Combinatorics.
16. (14 June 2012). "On Untouchable Numbers and Related Problems". [[Dartmouth College]].
17. Helfgott, Harald Andres. (2014). "Seoul [[International Congress of Mathematicians]] Proceedings". Kyung Moon SA.
18. Tao, Terence. (March 2014). "Every odd number greater than 1 has a representation is the sum of at most five primes". Mathematics of Computation.
19. Burnstein, Michael. (1978). "Kuratowski-Pontrjagin theorem on planar graphs". [[Journal of Combinatorial Theory]].
20. Robert L. Griess, Jr.. (1998). "Twelve Sporadic Groups". Springer-Verlag.
21. (2008). "The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2". [[Elsevier]].
22. Wilson, Robert A.. (2009). "The odd local subgroups of the Monster". [[Cambridge University Press]].
23. 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
24. "Orotariko Euskal Hiztegia".
25. "PBS – Islam: Empire of Faith – Faith – Five Pillars".
26. Sarhangi, Reza. (2012). "Interlocking Star Polygons in Persian Architecture: The Special Case of the Decagram in Mosaic Designs". Nexus Network Journal.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::