5

${\displaystyle 5}$ is the third smallest prime number, and the second super-prime.[1] It is the first safe prime, the first good prime, and the first of three known Wilson primes.[2] Five is the second Fermat prime[1] and the third Mersenne prime exponent,[3] as well as the third Catalan number,[4] and the third Sophie Germain prime.[1] Notably, 5 is equal to the sum of the only consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7). It is also a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial,[5] and an Eisenstein prime with no imaginary part and real part of the form ${\displaystyle 3p}$ − ${\displaystyle 1}$.[1] Five is the second square pyramidal number, and in particular, the first congruent number, since 5 is the length of the hypotenuse of the smallest integer-sided right triangle.[6]

Five is the second Fermat prime of the form ${\displaystyle 2^{2^{n}}}$+ ${\displaystyle 1}$, and more generally the second Sierpiński number of the first kind, ${\displaystyle n^{n}}$+ ${\displaystyle 1}$.[7] There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.[8] The sum of the first 3 Fermat primes, 3, 5 and 17, yields 25 (or 5²), while 257 is the 55th prime number. Combinations from these 5 Fermat primes generate 31 polygons with an odd number of sides that can be construncted purely with a compass and straight-edge, which includes the regular pentagon. Figurate numbers representing pentagons are called pentagonal numbers.

The number 5 is the fifth Fibonacci number, being 2 plus 3.[1] It is the only Fibonacci number that is equal to its position aside from 1, which is both the first and second Fibonacci numbers. Five is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... (OEIS: A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers.[9]

5 is the third Mersenne prime exponent of the form ${\displaystyle 2^{n}}$ − ${\displaystyle 1}$, which yields 31, in-turn the third exponent of one of only four known double Mersenne primes, 2,147,483,647. A fifth candidate double Mersenne prime is ${\displaystyle M_{M_{61}}}$ = 2^{23058...93951} − 1, which is too large to compute with current computers. In a related sequence, the first 5 terms in the sequence of Catalan–Mersenne numbers ${\displaystyle M_{c_{n}}}$ are the only known prime terms. A sixth possible candidate ${\displaystyle c_{5}}$ ~ 10^1037.7094 is also too large to compute for the time being. Double Mersenne primes as well as the Catalan-Mersenne primes are conjectured to be prime up to a certain limit.

Five is conjectured to be the only odd untouchable number,[10] and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.

5 is the value of the central cell of all 8 equivalent order-3 normal magic squares,[11] and of the only non-trivial (order-3) normal magic hexagon that is made of 19 cells.[12]

While polynomial equations of degree 4 and below can be solved with radicals, quintic equations of degree 5, and higher, cannot generally be so solved. This is the Abel–Ruffini theorem. This is related to the fact that the symmetric group ${\displaystyle \mathrm {S} _{n}}$ is a solvable group for n ≤ 4 and not solvable for n ≥ 5.

The sums of the first ${\displaystyle 5}$ non-primes greater than zero ${\displaystyle 1}$ + ${\displaystyle 4}$ + ${\displaystyle 6}$ + ${\displaystyle 8}$ + ${\displaystyle 9}$ and the first ${\displaystyle 5}$ prime numbers ${\displaystyle 2}$ + ${\displaystyle 3}$ + ${\displaystyle 5}$ + ${\displaystyle 7}$ + ${\displaystyle 11}$ both equal ${\displaystyle 28}$; the ${\displaystyle 7^{th}}$ triangular number and like ${\displaystyle 6}$ a perfect number, which also includes ${\displaystyle 496}$, the ${\displaystyle 31^{st}}$ triangular number and perfect number of the form ${\displaystyle 2^{p}}$⁻¹(${\displaystyle 2^{p}}$ − ${\displaystyle 1}$) with a ${\displaystyle p}$ of ${\displaystyle 5}$, by the Euclid–Euler theorem.[13][14][15]

There are five known unitary perfect numbers. A sixth unitary number, if discovered, would have at least nine odd prime factors.[16]

A pentagram is a five-pointed polygram, and the first proper star polygon, constructed from the diagonals of a regular pentagon as self-intersecting edges that are proportioned in golden ratio, ${\displaystyle \varphi }$. Its internal geometry appears prominently in Penrose tilings, and is a facet inside Kepler-Poinsot star polyhedra and Schläfli–Hess star polychora, represented by its Schläfli symbol {5/2}. A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges. In general, star polytopes that are regular only exist in dimensions 2 ≤ ${\displaystyle n}$ < 5.

Euler's identity, ${\displaystyle e^{i\pi }}$+ ${\displaystyle 1}$ = ${\displaystyle 0}$, contains five essential numbers used widely in mathematics: Archimedes' constant ${\displaystyle \pi }$, Euler's number ${\displaystyle e}$, the imaginary number ${\displaystyle i}$, unity ${\displaystyle 1}$, and zero ${\displaystyle 0}$, which makes this formula one of the most renown examples of mathematical beauty.

While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar: K₅, the complete graph with 5 vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram.[17] A similar graph is the Petersen graph, which is strongly connected and also nonplanar. It is most easily described as graph of a pentagram embedded inside a pentagon, with a total of 5 crossings, a girth of 5, and a Thue number of 5.[18][19] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.[20] The automorphism group of the Petersen graph is the symmetric group ${\displaystyle \mathrm {S} _{5}}$, of order 120 = 5!.

There are five Platonic solids,[21] five regular uniform polyhedron compounds,[22] and five regular abstract polyhedra.[23] There are also five space-filling convex polyhedra,[24] and five cell-transitive space-filling parallelohedra.[25]

The 5-cell, or pentatope, is the fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry ${\displaystyle \mathrm {A} _{4}}$ = 120 = 5! and ${\displaystyle \mathrm {S} _{5}}$ group structure, as with the Petersen graph. Its Petrie polygon is a regular pentagon, and its orthogonal projection is equivalent to the complete graph K₅. It is one of six regular 4-polytopes, made of 31 elements: 5 vertices, 10 edges, 10 faces, 5 tetrahedral cells and 1 4-face. 120 5-cells fit inside a 120-cell, the dual polychoron to the 600-cell, both regular 4-polytopes. Also, 5 24-cells fit inside a small stellated 120-cell, the first stellation of the 120-cell. A subset of the vertices of the small stellated 120-cell are matched by the great duoantiprism star, which is the only uniform nonconvex duoantiprismatic solution in the fourth dimension, constructed from the polytope cartesian product {5}⊗{5/3} and made of 10 pentagrammic crossed antiprisms, 50 tetrahedra, and 10 pentagonal antiprisms. On the other hand, the 5-simplex is the five-dimensional analogue of the 5-cell, or 4-simplex; the fifth iteration of ${\displaystyle n}$-simplexes in any ${\displaystyle n}$ dimensions. The 5-simplex has the Coxeter group ${\displaystyle \mathrm {A} _{5}}$ as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group ${\displaystyle \mathrm {S} _{6}}$, the only finite symmetric group which has an outer automorphism.

Analogues of three-dimensional icosahedral symmetry ${\displaystyle \mathrm {I} _{h}}$ (or two-dimensional pentagonal dihedral ${\displaystyle \mathrm {D} _{5}}$ symmetry, and four-dimensional ${\displaystyle \mathrm {H} _{4}}$ symmetry) do not exist in dimensions n ≥ 5. However, there is the uniform polychoral prismatic group ${\displaystyle \mathrm {H} _{4}}$ × ${\displaystyle \mathrm {A} _{1}}$ in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have ${\displaystyle \mathrm {H} _{4}}$ symmetry.

The highest dimension for compact hyperbolic honeycombs is the fifth dimension, with paracompact hyperbolic solutions existing up through 10-space. In the fifth dimension, there are five regular paracompact honeycombs, all with infinite facets and vertex figures.[26]

The five Mathieu groups constitute the first generation in the happy family of sporadic groups. These are also the first five sporadic groups to have been described, defined as ${\displaystyle \mathrm {M} _{n}}$ multiply transitive permutation groups on ${\displaystyle n}$ objects, with ${\displaystyle n}$ ∈ {11, 12, 22, 23, 24}. In particular, ${\displaystyle \mathrm {M} _{11}}$, the smallest of all sporadic groups, has a rank 3 action on 55 points from an induced action on unordered pairs, as well as two five-dimensional faithful complex irreducible representations over the field with 3 elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with n elements.[27] Of precisely 5 different conjugacy classes of maximal subgroups of ${\displaystyle \mathrm {M} _{11}}$, one is the almost simple symmetric group ${\displaystyle \mathrm {S} _{5}}$ (of order 5!), and another is ${\displaystyle \mathrm {M} _{10}}$, also almost simple, that functions as a point stabilizer which has 5 as its largest prime factor in its group order: 2⁴·3²·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas ${\displaystyle \mathrm {M} _{11}}$ is sharply 4-transitive, ${\displaystyle \mathrm {M} _{12}}$ is sharply 5-transitive and ${\displaystyle \mathrm {M} _{24}}$ is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups. ${\displaystyle \mathrm {M} _{22}}$ has the first five prime numbers as its distinct prime factors in its order of 2⁷·3²·5·7·11, and is the smallest of 5 sporadic groups with 5 distinct prime factors in their order. All Mathieu groups are subgroups of ${\displaystyle \mathrm {M} _{24}}$, which under the Witt design ${\displaystyle \mathrm {W} _{24}}$ of Steiner system S(5, 8, 24) emerges a construction of the extended binary Golay code ${\displaystyle \mathrm {B} _{24}}$ that has ${\displaystyle \mathrm {M} _{24}}$ as its automorphism group. ${\displaystyle \mathrm {W} _{24}}$ generates octads from code words of Hamming weight 8 from the extended binary Golay code, one of 5 different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24. The Witt design and the extended binary Golay code in turn can be used to generate a faithful construction of the 24-dimensional Leech lattice Λ₂₄, which is the subject of the second generation of seven sporadic groups that are subquotients of the automorphism of the Leech lattice, Conway group ${\displaystyle \mathrm {Co} _{0}}$.

There are five solvable groups that are excluded from finite simple groups of Lie type, and more importantly, five exceptional Lie groups: ${\displaystyle {\mathfrak {g}}_{2}}$, ${\displaystyle {\mathfrak {f}}_{4}}$, ${\displaystyle {\mathfrak {e}}_{6}}$, ${\displaystyle {\mathfrak {e}}_{7}}$, and ${\displaystyle {\mathfrak {e}}_{8}}$.

There are five non-supersingular primes: 37, 43, 53, 61, and 67, all smaller than the largest supersingular prime divisor of the friendly giant, 71.[28] All primes greater than 71 are non-supersingular.

The evolution of the modern Western digit for the numeral 5 cannot be traced back to the Indian system, as for the digits 1 to 4. The Kushana and Gupta empires in what is now India had among themselves several different forms that bear no resemblance to the modern digit. The Nagari and Punjabi took these digits and all came up with forms that were similar to a lowercase "h" rotated 180°. The Ghubar Arabs transformed the digit in several different ways, producing from that were more similar to the digits 4 or 3 than to 5.[29] It was from those digits that Europeans finally came up with the modern 5.

While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in .

On the seven-segment display of a calculator, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa.

• The atomic number of boron.[30]
• The number of appendages on most starfish, which exhibit pentamerism.[31]
• The most destructive known hurricanes rate as Category 5 on the Saffir–Simpson hurricane wind scale.[32]
• The most destructive known tornadoes rate an F-5 on the Fujita scale or EF-5 on the Enhanced Fujita scale.[33]

• 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).[89]
• In AFL Women's, the top level of women's Australian rules football, each team is allowed 5 "interchanges" (substitute players), who can be freely substituted at any time.
• In baseball scorekeeping, the number 5 represents the third baseman's position.
• In basketball:
  • The number 5 is used to represent the position of center.
  • Each team has five players on the court at a given time. Thus, the phrase "five on five" is commonly used to describe standard competitive basketball.[90]
  • The "5-second rule" refers to several related rules designed to promote continuous play. In all cases, violation of the rule results in a turnover.
  • Under the FIBA (used for all international play, and most non-US leagues) and NCAA women's rule sets, a team begins shooting bonus free throws once its opponent has committed five personal fouls in a quarter.
  • Under the FIBA rules, A player fouls out and must leave the game after committing five fouls
• Five-a-side football is a variation of association football in which each team fields five players.[91]
• In ice hockey:
  • A major penalty lasts five minutes.[92]
  • There are five different ways that a player can score a goal (teams at even strength, team on the power play, team playing shorthanded, penalty shot, and empty net).[93]
  • The area between the goaltender's legs is known as the five-hole.[94]
• In most rugby league competitions, the starting left wing wears this number. An exception is the Super League, which uses static squad numbering.
• In rugby union:
  • A try is worth 5 points.[95]
  • One of the two starting lock forwards wears number 5, and usually jumps at number 4 in the line-out.
  • In the French variation of the bonus points system, a bonus point in the league standings is awarded to a team that loses by 5 or fewer points.

5 as a resin identification code, used in recycling.

• 5 is the most common number of gears for automobiles with manual transmission.[96]
• In radio communication, the term "Five by five" is used to indicate perfect signal strength and clarity.[97]
• On almost all devices with a numeric keypad such as telephones, computers, etc., the 5 key has a raised dot or raised bar to make dialing easier. Persons who are blind or have low vision find it useful to be able to feel the keys of a telephone. All other numbers can be found with their relative position around the 5 button (on computer keyboards, the 5 key of the numpad has the raised dot or bar, but the 5 key that shifts with % does not).[98]
• On most telephones, the 5 key is associated with the letters J, K, and L,[99] but on some of the BlackBerry phones, it is the key for G and H.
• The Pentium, coined by Intel Corporation, is a fifth-generation x86 architecture microprocessor.[100]
• The resin identification code used in recycling to identify polypropylene.[101]

International maritime signal flag for 5

St. Petersburg Metro, Line 5

The fives of all four suits in playing cards

Five can refer to:

• "Give me five" is a common phrase used preceding a high five.
• An informal term for the British Security Service, MI5.
• Five babies born at one time are quintuplets. The most famous set of quintuplets were the Dionne quintuplets born in the 1930s.[102]
• In the United States legal system, the Fifth Amendment to the United States Constitution can be referred to in court as "pleading the fifth", absolving the defendant from self-incrimination.[103]
• Pentameter is verse with five repeating feet per line; iambic pentameter was the most popular form in Shakespeare.[104]
• Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth)[105]
• The designation of an Interstate Highway (Interstate 5) that runs from San Diego, California to Blaine, Washington.[106] In addition, all major north-south Interstate Highways in the United States end in 5.[107]
• In the computer game Riven, 5 is considered a holy number, and is a recurring theme throughout the game, appearing in hundreds of places, from the number of islands in the game to the number of bolts on pieces of machinery.
• The Garden of Cyrus (1658) by Sir Thomas Browne is a Pythagorean discourse based upon the number 5.
• The holy number of Discordianism, as dictated by the Law of Fives.[108]
• The number of Justices on the Supreme Court of the United States necessary to render a majority decision.[109]
• The number of dots in a quincunx.[110]
• The number of permanent members with veto power on the United Nations Security Council.[111]
• The number of sides and the number of angles in a pentagon.[112]
• The number of points in a pentagram.[113]
• The number of Korotkoff sounds when measuring blood pressure[114]
• The drink Five Alive is named for its five ingredients. The drink punch derives its name after the Sanskrit पञ्च (pañc) for having five ingredients.[115]
• The Keating Five were five United States Senators accused of corruption in 1989.[116]
• The Inferior Five: Merryman, Awkwardman, The Blimp, White Feather, and Dumb Bunny. DC Comics parody superhero team.[117]
• No. 5 is the name of the iconic fragrance created by Coco Chanel.[118]
• The Committee of Five was delegated to draft the United States Declaration of Independence.[119]
• The five-second rule is a commonly used rule of thumb for dropped food.[120]
• 555 95472, usually referred to simply as 5, is a minor male character in the comic strip Peanuts.[121]

• Five Families
• Five Nations (disambiguation)
• 555 (number)
• List of highways numbered 5

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 58–67

• Media related to 5 (number) at Wikimedia Commons
• The dictionary definition of five at Wiktionary
• The Number 5
• The Positive Integer 5
• Prime curiosities: 5