 

Cardinal  five  
Ordinal  5th (fifth)  
Numeral system  quinary  
Factorization  prime  
Prime  3rd  
Divisors  1, 5  
Greek numeral  Ε´  
Roman numeral  V, v  
Greek prefix  penta/pent  
Latin prefix  quinque/quinqu/quint  
Binary  101_{2}  
Ternary  12_{3}  
Senary  5_{6}  
Octal  5_{8}  
Duodecimal  5_{12}  
Hexadecimal  5_{16}  
Greek  ε (or Ε)  
Arabic, Kurdish  ٥  
Persian, Sindhi, Urdu  ۵  
Ge'ez  ፭  
Bengali  ৫  
Kannada  ೫  
Punjabi  ੫  
Chinese numeral  五  
Armenian  Ե  
Devanāgarī  ५  
Hebrew  ה  
Khmer  ៥  
Telugu  ౫  
Malayalam  ൫  
Tamil  ௫  
Thai  ๕  
Babylonian numeral  𒐙  
Egyptian hieroglyph, Chinese counting rod    
Maya numerals  𝋥  
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.
Five is the thirdsmallest prime number,^{[1]} equal to the sum of the only consecutive positive integers to also be prime numbers (2 + 3). In integer sequences, five is also the second Fermat prime, and the third Mersenne prime exponent, as well as the fourth or fifth Fibonacci number;^{[2]} 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integersided right triangle, making part of the smallest Pythagorean triple (3, 4, 5).^{[3]}
In geometry, the regular fivesided pentagon is the first regular polygon that does not tile the plane with copies of itself, and it is the largest face that any of the five regular threedimensional regular Platonic solid can have, as represented in the regular dodecahedron. For curves, a conic is determined using five points in the same way that two points are needed to determine a line.^{[4]}
In abstract algebra and the classification of finite simple groups, five is the count of exceptional Lie groups as well as the number of Mathieu groups that are sporadic groups. Five is also, more elementarily, the number of properties that are used to distinguish between the four fundamental number systems used in mathematics, which are rooted in the real numbers.
Historically, 5 has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
In the classification of number systems, the real numbers and its three subsequent Cayley–Dickson constructions of algebras over the field of the real numbers (i.e. the complex numbers , the quaternions , and the octonions ) are normed division algebras that hold up to five different principal algebraic properties of interest: whether the algebras are ordered, and whether they hold commutative, associative, alternative, and powerassociative multiplicative properties.^{[5]} Whereas the real numbers contain all five properties, the octonions are only alternative and powerassociative. In comparison, the sedenions , which represent a fifth algebra in this series, is not a composition algebra unlike and , is only powerassociative, and is the first algebra to contain nontrivial zero divisors as with all further algebras over larger fields.^{[6]} Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16.
Five is the third prime number, and more specifically, the second superprime since its prime index is prime.^{[1]} Aside from being the sum of the only consecutive positive integers to also be prime numbers, 2 + 3, it is also the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7);^{[7]}^{[8]} this makes it the first balanced prime with equalsized prime gaps above and below it (of 2).^{[9]} 5 is the first safe prime^{[10]} where for a prime is also prime (2), and the first good prime, since it is the first prime number whose square (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 and 11 × 2 = 22 are less than 25).^{[11]} 11, the fifth prime number, is the next good prime, that also forms the first pair of sexy primes with 5.^{[12]} More significantly, the fifth Heegner number that forms an imaginary quadratic field with unique factorization is also 11^{[13]} (and the first repunit prime in decimal, a base inwhich five is also the first nontrivial 1automorphic number).^{[14]} 5 is also an Eisenstein prime (like 11) with no imaginary part and real part of the form .^{[1]}
5 is the first prime number (and more generally, natural number) that is palindromic for a base where , with adjacent numbers 4 and 6 the only two composite numbers to be strictly nonpalindromic in such sense.^{[15]} In other words, all numbers greater than 6 in this sequence are prime, where 11 is the next strictly nonpalindromic number after 6, equal to the sum of all nonprime entries in the sequence (0, 1, 4, 6). Positive integers have representations as sums of three palindromic numbers only in bases greater than or equal to five (quinary).^{[16]}
All prime numbers greater than or equal to 5 are congruent to (as well as, ).
5 is the third Mersenne prime exponent for , which yields the eleventh prime number and fifth superprime 31.^{[17]}^{[18]}^{[1]} This is the prime index of the third Mersenne prime and second double Mersenne prime 127,^{[19]} as well as the third double Mersenne prime exponent for the number 2,147,483,647,^{[19]} which is the largest value that a signed 32bit integer field can hold. Collectively, 5 and 31 generate a sum of 36 (the square of 6) and a difference of 26, which is the only number to lie between a square and a cube (respectively, 25 and 27).^{[20]}
There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime = 2^{23058...93951} − 1 too large to compute with current computers. In a related sequence, the first five terms in the sequence of Catalan–Mersenne numbers are the only known prime terms, with a sixth possible candidate in the order of 10^{1037.7094}. These prime sequences are conjectured to be prime up to a certain limit.
5 is the second Fermat prime of the form , and more generally the second Sierpiński number of the first kind, .^{[21]} There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.^{[22]} The sum of the first three Fermat primes, 3, 5 and 17, yields 25 or 5^{2}, while 257 is the 55th prime number. Combinations from these five Fermat primes generate thirtyone polygons with an odd number of sides that can be constructed purely with a compass and straightedge, which includes the fivesided regular pentagon.^{[23]}^{[24]}^{: pp.137–142 } Apropos, thirtyone is also equal to the sum of the maximum number of areas inside a circle that are formed from the sides and diagonals of the first five sided polygons, which is equal to the maximum number of areas formed by a sixsided polygon; per Moser's circle problem.^{[25]}^{[24]}^{: pp.76–78 }
5 is also the first of three known Wilson primes (5, 13, 563),^{[26]} where the square of a prime divides In the case of ,
The first two Wilson primes are also consecutive Proth primes^{[27]} and Markov numbers, where 5 appears in solutions to the Markov Diophantine equations: (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). 5 is also the third factorial prime,^{[28]} since , and the first nontrivial alternating factorial equal to the absolute value of the alternating sum of the first three factorials, ^{[29]}
The sums of the first five nonprimes greater than zero 1 + 4 + 6 + 8 + 9 and the first five prime numbers 2 + 3 + 5 + 7 + 11 both equal 28; the seventh triangular number and like 6 a perfect number, which also includes 496, the thirtyfirst triangular number and perfect number of the form () with a of , by the Euclid–Euler theorem.^{[30]}^{[31]}^{[32]} Within the larger family of Ore numbers, 140 and 496, respectively the fourth and sixth indexed members, both contain a set of divisors that produce integer harmonic means equal to 5.^{[33]}^{[34]} The fifth Mersenne prime, 8191,^{[18]} splits into 4095 and 4096, with the latter being the fifth superperfect number^{[35]} and the sixth power of four, 4^{6}.
Five is also the total number of known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors.^{[36]}^{[37]} The smallest such number is 6, and the largest of these is equivalent to the sum of 4095 divisors, where 4095 is the largest of five Ramanujan–Nagell numbers that are both triangular numbers and Mersenne numbers of the general form.^{[38]}^{[39]}
The factorial of five is multiply perfect like 28 and 496.^{[40]} It is the sum of the first fifteen nonzero positive integers and 15th triangular number, which inturn is the sum of the first five nonzero positive integers and 5th triangular number. Furthermore, , where 125 is the second number to have an aliquot sum of 31 (after the fifth power of two, 32).^{[41]}
5 is a pentagonal number in the sequence of figurate numbers, which starts: 1, 5, 12, 22, 35, ...^{[42]}
31 is the first prime centered pentagonal number,^{[47]} and the fifth centered triangular number.^{[48]} The fifth pentagonal and tetrahedral number is 35, which is equal to the sum of the first five triangular numbers: 1, 3, 6, 10, 15.^{[49]} In the sequence of pentatope numbers that start from the first (or fifth) cell of the fifth row of Pascal's triangle (left to right or from right to left), the first few terms are: 1, 5, 15, 35, 70, 126, 210, 330, 495, ...^{[50]} The first five members in this sequence add to 126, which is also the sixth pentagonal pyramidal number^{[51]} as well as the fifth perfect Granville number.^{[52]} This is the third Granville number not to be perfect, and the only known such number with three distinct prime factors.^{[53]}
55 is the fifteenth discrete biprime,^{[54]} equal to the product between 5 and the fifth prime and third superprime 11.^{[1]} These two numbers also form the second pair (5, 11) of Brown numbers such that where five is also the second number that belongs to the first pair (4, 5); altogether only five distinct numbers (4, 5, 7, 11, and 71) are needed to generate the set of known pairs of Brown numbers, where the third and largest pair is (7, 71).^{[55]}^{[56]} Fiftyfive is also the tenth Fibonacci number,^{[57]} whose digit sum is also 10, in its decimal representation. It is the tenth triangular number and the fourth that is doubly triangular,^{[58]} the fifth heptagonal number^{[59]} and fourth centered nonagonal number,^{[60]} and as listed above, the fifth square pyramidal number.^{[44]} The sequence of triangular that are powers of 10 is: 55, 5050, 500500, ...^{[61]} 55 in baseten is also the fourth Kaprekar number as are all triangular numbers that are powers of ten, which initially includes 1, 9 and 45,^{[62]} with fortyfive itself the ninth triangular number where 5 lies midway between 1 and 9 in the sequence of natural numbers. 45 is also conjectured by Ramsey number ,^{[63]}^{[64]} and is a Schröder–Hipparchus number; the next and fifth such number is 197, the fortyfifth prime number^{[17]} that represents the number of ways of dissecting a heptagon into smaller polygons by inserting diagonals.^{[65]} A fivesided convex pentagon, on the other hand, has eleven ways of being subdivided in such manner.
Five 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.^{[66]}
Where five is the third prime number and odd number, 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^{[67]} (also known as the odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes peerreview. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).^{[68]}
As a consequence of Fermat's little theorem and Euler's criterion, all squares are congruent to 0, 1, 4 (or −1) modulo 5.^{[69]} All integers can be expressed as the sum of five nonzero squares.^{[70]}^{[71]}
Regarding Waring's problem, , where every natural number is the sum of at most thirtyseven fifth powers.^{[72]}^{[73]}
In the Collatz 3x + 1 problem, 5 requires five steps to reach one by multiplying terms by three and adding one if the term is odd (starting with five itself), and dividing by two if they are even: {5 ➙ 16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}; the only other number to require five steps is 32 since 16 must be part of such path (see image to the right for a map of orbits for small odd numbers).^{[74]}^{[75]}
Specifically, 120 needs fifteen steps to arrive at 5: {120 ➙ 60 ➙ 30 ➙ 15 ➙ 46 ➙ 23 ➙ 70 ➙ 35 ➙ 106 ➙ 53 ➙ 160 ➙ 80 ➙ 40 ➙ 20 ➙ 10 ➙ 5}. These comprise a total of sixteen numbers before cycling through {16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}, where 16 is the smallest number with exactly five divisors,^{[76]} and one of only two numbers to have an aliquot sum of 15, the other being 33.^{[41]} Otherwise, the trajectory of 15 requires seventeen steps to reach 1,^{[75]} where its reduced Collatz trajectory is equal to five when counting the steps {23, 53, 5, 2, 1} that are prime, including 1.^{[77]} Overall, thirteen numbers in the Collatz map for 15 back to 1 are composite,^{[74]} where the largest prime in the trajectory of 120 back to {4 ➙ 2 ➙ 1 ➙ 4 ➙ ...} is the sixteenth prime number, 53.^{[17]}
When generalizing the Collatz conjecture to all positive or negative integers, −5 becomes one of only four known possible cycle starting points and endpoints, and in its case in five steps too: {−5 ➙ −14 ➙ −7 ➙ −20 ➙ −10 ➙ −5 ➙ ...}. The other possible cycles begin and end at −17 in eighteen steps, −1 in two steps, and 1 in three steps. This behavior is analogous to the path cycle of five in the 3x − 1 problem, where 5 takes five steps to return cyclically, in this instance by multiplying terms by three and subtracting 1 if the terms are odd, and also halving if even.^{[78]} It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).^{[79]}
In the Fibonacci sequence, which can be defined in terms of the golden ratio (see for example, Binet's formula), 5 is strictly the fifth Fibonacci number (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...) — being the sum of 2 and 3 — ^{[1]} as the only Fibonacci number greater than 1 that is equal to its position. In planar geometry, the ratio of a side and diagonal of a regular fivesided pentagon is also . Similarly, 5 is a member of the Perrin sequence, where 5 is both the fifth and sixth Perrin numbers, following (2, 3, 2) and preceding (7, 17);^{[80]} this sequence is instead associated with the plastic ratio, the least "small" Pisot–Vijayaraghavan number that does not supersede the golden ratio.^{[81]} This ratio is also associated with the Padovan sequence (1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, ...) where 5 is the twelfth member (and 12 the fifteenth), inwhich the −th Padovan number satisfies and ^{[82]} Manipulating Narayana's cows sequence that has relations in proportion with the supergolden ratio as the fourthsmallest PisotVijayaraghavan number whose value is less than the golden ratio, such that , five appears as the fourth member: (1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, ...).^{[83]}^{[84]} On the other hand, 5 is part of the sequence of Pell numbers as the third indexed member, (0, 1, 2, 5, 12, 29, 70, 169, 408, ...).^{[85]} These numbers are approximately proportional to powers of the secondsmallest Pisot Vijayaraghavan number following , the silver ratio (and analogous to Fibonacci numbers, as powers of ), that appears in the regular octagon.
There are five countably infinite Ramsey classes of permutations, where the age of each countable homogeneous permutation forms an individual Ramsey class of objects such that, for each natural number and each choice of objects , there is no object where in any coloring of all subobjects of isomorphic to there exists a monochromatic subobject isomorphic to .^{[86]}^{: pp.1, 2 } Aside from , the five classes of Ramsey permutations are the classes of:^{[86]}^{: p.4 }
In general, the Fraïssé limit of a class of finite relational structure is the age of a countable homogeneous relational structure if and only if five conditions hold for : it is closed under isomorphism, it has only countably many isomorphism classes, it is hereditary, it is jointembedded, and it holds the amalgamation property.^{[86]}^{: p.3 }
5 is the value of the central cell of the first nontrivial normal magic square, called the Luoshu square. Its array has a magic constant of , where the sums of its rows, columns, and diagonals are all equal to fifteen.^{[87]} On the other hand, a normal magic square^{[a]} has a magic constant of , where 5 and 13 are the first two Wilson primes.^{[26]} The fifth number to return for the Mertens function is 65,^{[88]} with counting the number of squarefree integers up to with an even number of prime factors, minus the count of numbers with an odd number of prime factors. 65 is the nineteenth biprime with distinct prime factors,^{[54]} with an aliquot sum of 19 as well^{[41]} and equivalent to 1^{5} + 2^{4} + 3^{3} + 4^{2} + 5^{1}.^{[89]} It is also the magic constant of the Queens Problem for ,^{[90]} the fifth octagonal number,^{[91]} and the Stirling number of the second kind that represents sixtyfive ways of dividing a set of six objects into four nonempty subsets.^{[92]} 13 and 5 are also the fourth and third Markov numbers, respectively, where the sixth member in this sequence (34) is the magic constant of a normal magic octagram and magic square.^{[93]} In between these three Markov numbers is the tenth prime number 29^{[17]} that represents the number of pentacubes when reflections are considered distinct; this number is also the fifth Lucas prime after 11 and 7 (where the first prime that is not a Lucas prime is 5, followed by 13).^{[94]} A magic constant of 505 is generated by a normal magic square,^{[93]} where 10 is the fifth composite.^{[95]}
5 is also the value of the central cell the only nontrivial normal magic hexagon made of nineteen cells.^{[96]}^{[b]} Where the sum between the magic constants of this order3 normal magic hexagon (38) and the order5 normal magic square (65) is 103 — the prime index of the third Wilson prime 563 equal to the sum of all three pairs of Brown numbers — their difference is 27, itself the prime index of 103.^{[17]} In baseten, 15 and 27 are the only twodigit numbers that are equal to the sum between their digits (inclusive, i.e. 2 + 3 + ... + 7 = 27), with these two numbers consecutive perfect totient numbers after 3 and 9.^{[97]} 103 is the fifth irregular prime^{[98]} that divides the numerator (236364091) of the twentyfourth Bernoulli number , and as such it is part of the eighth irregular pair (103, 24).^{[99]} In a twodimensional array, the number of planar partitions with a sum of four is equal to thirteen and the number of such partitions with a sum of five is twentyfour,^{[100]} a value equal to the sumofdivisors of the ninth arithmetic number 15^{[101]} whose divisors also produce an integer arithmetic mean of 6^{[102]} (alongside an aliquot sum of 9).^{[41]} The smallest value that the magic constant of a fivepointed magic pentagram can have using distinct integers is 24.^{[103]}^{[c]}
A pentagram, or fivepointed polygram, is the first proper star polygon constructed from the diagonals of a regular pentagon as selfintersecting edges that are proportioned in golden ratio, . Where the equilateral triangle is the first proper regular polygon and only polygon without diagonals, the regular pentagon contains the same number of edges and diagonals. Five is the sum of differences in the number of diagonals and sides of the first two regular polygons (which includes the square).^{[104]}
The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol {5/2}) appears prominently in Penrose tilings, and they are facets inside Kepler–Poinsot star polyhedra and Schläfli–Hess star polychora. A similar figure to the pentagram is a fivepointed simple isotoxal star ☆ without selfintersecting edges, often found inside Islamic Girih tiles (there are five different rudimentary types).^{[105]}
In graph theory, all graphs with four or fewer vertices are planar, however, there is a graph with five vertices that is not: K_{5}, the complete graph with five vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram. By Kuratowski's theorem, a finite graph is planar iff it does not contain a subgraph that is a subdivision of K_{5}, or the complete bipartite utility graph K_{3,3}.^{[106]} 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.^{[107]}^{[108]} The Petersen graph, which is also a distanceregular graph, is one of only 5 known connected vertextransitive graphs with no Hamiltonian cycles.^{[109]} The automorphism group of the Petersen graph is the symmetric group of order 120 = 5!. For polynomial equations of degree 4 and below can be solved with radicals, quintic equations of degree 5 and higher cannot generally be so solved (see, Abel–Ruffini theorem). This is related to the fact that the symmetric group is a solvable group for ⩽ , and not for ⩾ .
The chromatic number of the plane is at least five, depending on the choice of settheoretical axioms: the minimum number of colors required to color the plane such that no pair of points at a distance of 1 has the same color.^{[110]}^{[111]} Whereas the hexagonal Golomb graph and the regular hexagonal tiling generate chromatic numbers of 4 and 7, respectively, a chromatic coloring of 5 can be attained under a more complicated graph where multiple fourcoloring Moser spindles are linked so that no monochromatic triples exist in any coloring of the overall graph, as that would generate an equilateral arrangement that tends toward a purely hexagonal structure.
The plane also contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations: hexagonal, oblique, rectangular, centered rectangular, and square lattices. Uniform tilings of the plane, furthermore, are generated from combinations of only five regular polygons: the triangle, square, hexagon, octagon, and the dodecagon.^{[112]} The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.^{[113]}
There are five Platonic solids in threedimensional space that are regular: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.^{[114]} The dodecahedron in particular contains pentagonal faces, while the icosahedron, its 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, which are called the Archimedean solids. There are also five:
Moreover, the fifth pentagonal pyramidal number represents the total number of indexed uniform compound polyhedra,^{[115]} which includes seven families of prisms and antiprisms. Seventyfive is also the number of nonprismatic uniform polyhedra, which includes Platonic solids, Archimedean solids, and star polyhedra; there are also precisely five uniform prisms and antiprisms that contain pentagons or pentagrams as faces — the pentagonal prism and antiprism, and the pentagrammic prism, antiprism, and crossedantiprism.^{[122]} In all, there are twentyfive uniform polyhedra that generate fourdimensional uniform polychora, they are the five Platonic solids, fifteen Archimedean solids counting two enantiomorphic forms, and five associated prisms: the triangular, pentagonal, hexagonal, octagonal, and decagonal prisms.
The pentatope, or 5cell, is the selfdual fourthdimensional analogue of the tetrahedron, with Coxeter group symmetry of order 120 = 5! and group structure. Made of five tetrahedra, its Petrie polygon is a regular pentagon and its orthographic projection is equivalent to the complete graph K_{5}. It is one of six regular 4polytopes, made of thirtyone elements: five vertices, ten edges, ten faces, five tetrahedral cells and one 4face.^{[123]}^{: p.120 }
Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora based on only twentyfive uniform polyhedra: , , , , and , accompanied by a fifth or sixth general group of unique 4prisms of Platonic and Archimedean solids. There are also a total of five Coxeter groups that generate nonprismatic Euclidean honeycombs in 4space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite facets, as with the order5 5cell honeycomb and the order5 120cell honeycomb, both of which have five cells around each face. Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10. Likewise, analogues of fourdimensional hexadecachoric or icositetrachoric symmetry do not exist in dimensions ⩾ ; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4polytopes that have and symmetry. There are also five regular projective 4polytopes in the fourth dimension, all of which are hemipolytopes of the regular 4polytopes, with the exception of the 5cell.^{[127]} Only two regular projective polytopes exist in each higher dimensional space.
Generally, star polytopes that are regular only exist in dimensions ⩽ < , and can be constructed using five Miller rules for stellating polyhedra or higherdimensional polytopes.^{[128]}
In particular, Bring's surface is the curve in the projective plane that is represented by the homogeneous equations:^{[129]}
It holds the largest possible automorphism group of a genus four complex curve, with group structure . This is the Riemann surface associated with the small stellated dodecahedron, whose fundamental polygon is a regular hyperbolic icosagon, with an area of (by the GaussBonnet theorem). Including reflections, its full group of symmetries is , of order 240; which is also the number of (2,4,5) hyperbolic triangles that tessellate its fundamental polygon. Bring quintic holds roots that satisfy Bring's curve.
The 5simplex or hexateron is the fivedimensional analogue of the 5cell, or 4simplex. It has Coxeter group as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group , the only finite symmetric group which has an outer automorphism. The 5cube, made of ten tesseracts and the 5cell as its vertex figure, is also regular and one of thirtyone uniform 5polytopes under the Coxeter hypercubic group. The demipenteract, with one hundred and twenty cells, is the only fifthdimensional semiregular polytope, and has the rectified 5cell as its vertex figure, which is one of only three semiregular 4polytopes alongside the rectified 600cell and the snub 24cell. In the fifth dimension, there are five regular paracompact honeycombs, all with infinite facets and vertex figures; no other regular paracompact honeycombs exist in higher dimensions.^{[130]} There are also exclusively twelve complex aperiotopes in complex spaces of dimensions ⩾ ; alongside complex polytopes in and higher under simplex, hypercubic and orthoplex groups (with van Oss polytopes).^{[131]}
A Veronese surface in the projective plane generalizes a linear condition for a point to be contained inside a conic, where five points determine a conic.^{[4]}
There are five complex exceptional Lie algebras: , , , , and . The smallest of these, of real dimension 28, can be represented in fivedimensional complex space and projected as a ball rolling on top of another ball, whose motion is described in twodimensional space.^{[132]} is the largest, and holds the other four Lie algebras as subgroups, with a representation over in dimension 496. It contains an associated lattice that is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the 600cell, whose Euclidean norms define a quadratic form on a lattice structure isomorphic to the optimal configuration of spheres in eight dimensions.^{[133]} This sphere packing lattice structure in 8space is held by the vertex arrangement of the 5_{21} honeycomb, one of five Euclidean honeycombs that admit Gosset's original definition of a semiregular honeycomb, which includes the threedimensional alternated cubic honeycomb.^{[134]}^{[135]} The smallest simple isomorphism found inside finite simple Lie groups is ,^{[136]} where here represents alternating groups and classical Chevalley groups. In particular, the smallest nonsolvable group is the alternating group on five letters, which is also the smallest simple nonabelian group.
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 multiply transitive permutation groups on objects, with ∈ {11, 12, 22, 23, 24}.^{[137]}^{: p.54 } In particular, , the smallest of all sporadic groups, has a rank 3 action on fiftyfive points from an induced action on unordered pairs, as well as two fivedimensional faithful complex irreducible representations over the field with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with elements.^{[138]} Of precisely five different conjugacy classes of maximal subgroups of , one is the almost simple symmetric group (of order 5!), and another is , also almost simple, that functions as a point stabilizer which contains five as its largest prime factor in its group order: 2^{4}·3^{2}·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas is sharply 4transitive, is sharply 5transitive and is 5transitive, and as such they are the only two 5transitive groups that are not symmetric groups or alternating groups.^{[139]} has the first five prime numbers as its distinct prime factors in its order of 2^{7}·3^{2}·5·7·11, and is the smallest of five sporadic groups with five distinct prime factors in their order.^{[137]}^{: p.17 } All Mathieu groups are subgroups of , which under the Witt design of Steiner system emerges a construction of the extended binary Golay code that has as its automorphism group.^{[137]}^{: pp.39, 47, 55 } generates octads from code words of Hamming weight 8 from the extended binary Golay code, one of five different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24.^{[137]}^{: p.38 } The Witt design and the extended binary Golay code in turn can be used to generate a faithful construction of the 24dimensional Leech lattice Λ_{24}, which is primarily constructed using the Weyl vector that admits the only nonunitary solution to the cannonball problem, where the sum of the squares of the first twentyfour integers is equivalent to the square of another integer, the fifth pentatope number (70). The subquotients of the automorphism of the Leech lattice, Conway group , is in turn the subject of the second generation of seven sporadic groups.^{[137]}^{: pp.99, 125 }
There are five nonsupersingular prime numbers — 37, 43, 53, 61, and 67 — less than 71, which is the largest of fifteen supersingular primes that divide the order of the friendly giant, itself the largest sporadic group.^{[140]} In particular, a centralizer of an element of order 5 inside this group arises from the product between Harada–Norton sporadic group and a group of order 5.^{[141]}^{[142]} On its own, can be represented using standard generators that further dictate a condition where .^{[143]}^{[144]} This condition is also held by other generators that belong to the Tits group ,^{[145]} the only finite simple group that is a nonstrict group of Lie type that can also classify as sporadic (fifthlargest of all twentyseven by order, too). Furthermore, over the field with five elements, holds a 133dimensional representation where 5 acts on a commutative yet nonassociative product as a 5modular analogue of the Griess algebra ^{♮},^{[146]} which holds the friendly giant as its automorphism group.
Euler's identity, + = , contains five essential numbers used widely in mathematics: Archimedes' constant , Euler's number , the imaginary number , unity , and zero .^{[147]}^{[148]}^{[149]}
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  

5^{x}  5  25  125  625  3125  15625  78125  390625  1953125  9765625  48828125  244140625  1220703125  6103515625  30517578125  
x^{5}  1  32  243  1024  7776  16807  32768  59049  100000  161051  248832  371293  537824  759375 
All multiples of 5 will end in either 5 or 0, and vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions because they are prime factors of 10, the base.
In the powers of 5, every power ends with the number five, and from 5^{3} onward, if the exponent is odd, then the hundreds digit is 1, and if it is even, the hundreds digit is 6.
A number raised to the fifth power always ends in the same digit as .
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.^{[150]} 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 sevensegment 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 viceversa. It is one of three numbers, along with 4 and 6, where the number of segments matches the number.
There are five Lagrangian points in a twobody system.
There are usually considered to be five senses (in general terms); the five basic tastes are sweet, salty, sour, bitter, and umami.^{[151]} Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.^{[152]} Five is the number of appendages on most starfish, which exhibit pentamerism.^{[153]}
5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.^{[154]}
A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.^{[155]}
Modern musical notation uses a musical staff made of five horizontal lines.^{[156]} A scale with five notes per octave is called a pentatonic scale.^{[157]} A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.^{[158]} 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.
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").^{[159]} 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.^{[160]}
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).^{[161]}
The Five Pillars of Islam.^{[162]}
The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.
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 NeoPagan 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.^{[163]} 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.^{[164]} Also, the traditional Japanese calendar has a fiveday weekly cycle that can be still observed in printed mixed calendars combining Western, ChineseBuddhist, and Japanese names for each weekday. There are also five elements in the traditional Chinese Wuxing.^{[165]}
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.^{[166]} The pentagram, or fivepointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.
The only known number which can be expressed as the sum of the first nonnegative integers (1 + 2 + 3 + 4 + 5 + 6 + 7), the first primes (2 + 3 + 5 + 7 + 11) and the first nonprimes (1 + 4 + 6 + 8 + 9). There is probably no other number with this property.
A coloring of the set of edges of a graph G is called nonrepetitive if the sequence of colors on any path in G is nonrepetitive...In Fig. 1 we show a nonrepetitive 5coloring of the edges of P... Since, as can easily be checked, 4 colors do not suffice for this task, we have π(P) = 5.
There are five basic tastes: sweet, salty, sour, bitter and umami...
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 fivedigit stage.
The five appendages of the starfish are thought to be homologous to five human buds
5 5 005 ENQ (enquiry)
The most common accentualsyllabic lines are fivefoot iambic lines (iambic pentameter)
the five lines and four spaces between them on which musical notes are written
Pentatonic scales, as used in jazz, are five note scales
are the perfect fourth, perfect fifth, and the octave
The first category is the Five Agents [Elements] namely, Water, Fire, Wood, Metal, and Earth.
The Japanese names of the days of the week are taken from the names of the seven basic nature symbols
Plato and Aristotle postulated a fifth state of matter, which they called "idea" or quintessence" (from "quint" which means "fifth")
quincunx five points
