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 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 ways, producing from that were more similar to the digits 4 or 3 than to 5.[1] 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 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.
5 is the fifth
Fibonacci number, being 2 plus 3,[2] and the only Fibonacci number that is equal to its position aside from
1 (that is also the second index). 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). In the Perrin sequence 5 is both the fifth and sixth
Perrin numbers.[18]
5 is also the third
Mersenne prime exponent of the form , which yields , the eleventh prime number and fifth
super-prime.[24][25][2] This is the
prime index of the third
Mersenne prime and second
double Mersenne prime127,[26] as well as the third double Mersenne prime exponent for the number
2,147,483,647,[26] which is the largest value that a
signed32-bitinteger field can hold. There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime = 223058...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 101037.7094. These prime sequences are conjectured to be prime up to a certain limit.
There are a total of five known
unitary perfect numbers, which are numbers that are the sums of their positive
properunitary divisors.[27][28] 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.[29][30] The sums of the first five non-primes 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 thirty-first triangular number and perfect number of the form () with a of , by the
Euclid–Euler theorem.[31][32][33] 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.[34][35] The fifth Mersenne prime,
8191,[25] splits into 4095 and
4096, with the latter being the fifth
superperfect number[36] and the sixth power of four, 46.
The
factorial of five is
multiply perfect like 28 and 496.[42] It is the sum of the first
fifteen non-zero positive
integers and 15th
triangular number, which in-turn is the sum of the first five non-zero 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).[43] On its own, 31 is the first prime
centered pentagonal number,[44] and the fifth
centered triangular number.[45] Collectively, five and thirty-one 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).[46] 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.[47] 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, ...[48] The first five
members in this sequence add to
126, which is also the sixth
pentagonal pyramidal number[49] as well as the fifth -perfect
Granville number.[50] This is the third Granville number not to be perfect, and the only known such number with three distinct prime factors.[51]
55 is the fifteenth discrete
biprime,[52] equal to the product between 5 and the fifth prime and third super-prime
11.[2] 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).[53][54]
Fifty-five is also the tenth
Fibonacci number,[55] whose
digit sum is also
10, in its
decimal representation. It is the tenth triangular number and the fourth that is
doubly triangular,[56] the fifth
heptagonal number[57] and fourth
centered nonagonal number,[58] and as listed above, the fifth square pyramidal number.[39] The sequence of triangular that are
powers of 10 is: 55, 5050,
500500, ...[59] 55 in base-ten is also the fourth
Kaprekar number as are all triangular numbers that are powers of ten, which initially includes
1,
9 and
45,[60] with forty-five 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,[61][62] and is a
Schröder–Hipparchus number; the next and fifth such number is
197, the forty-fifth prime number[24] that represents the number of ways of dissecting a
heptagon into smaller polygons by inserting
diagonals.[63] A five-sided
convexpentagon, on the other hand, has eleven ways of being subdivided in such manner.[a]
Magic figures
5 is the value of the central
cell of the first non-trivial
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.[64] On the other hand, a normal magic square[b] has a magic constant of , where 5 and
13 are the first two
Wilson primes.[4] The fifth number to return for the
Mertens function is
65,[65] with counting the number of
square-free 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,[52] with an aliquot sum of
19 as well[43] and equivalent to 15 + 24 + 33 + 42 + 51.[66] It is also the magic constant of the Queens Problem for ,[67] the fifth
octagonal number,[68] and the
Stirling number of the second kind that represents sixty-five ways of dividing a
set of six objects into four non-empty
subsets.[69] 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.[70] In between these three Markov numbers is the tenth prime number
29[24] 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).[71] A magic constant of 505 is generated by a normal magic square,[70] where
10 is the fifth
composite.[72]
5 is also the value of the central cell the only non-trivial
normal magic hexagon made of nineteen cells.[73][c] Where the sum between the magic constants of this order-3 normal magic hexagon (
38) and the order-5 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.[24] In base-ten, 15 and
27 are the only two-digit 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.[74] 103 is the fifth
irregular prime[75] that divides the
numerator (236364091) of the twenty-fourth
Bernoulli number, and as such it is part of the eighth
irregular pair (103, 24).[76] In a two-dimensional 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 twenty-four,[77] a value equal to the
sum-of-divisors of the ninth
arithmetic number 15[78] whose divisors also produce an integer
arithmetic mean of
6[79] (alongside an
aliquot sum of 9).[43] The smallest value that the magic constant of a five-pointed
magic pentagram can have using distinct integers is 24.[80][d]
Collatz conjecture
In the
Collatz3x + 1problem, 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 [e] for a map of
orbits for small odd numbers).[81][82]
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,[83] and one of only two numbers to have an aliquot sum of
15, the other being
33.[43] Otherwise, the trajectory of 15 requires seventeen steps to reach 1,[82] where its reduced Collatz trajectory is equal to five when counting the steps {23, 53, 5, 2, 1} that are prime, including 1.[84] Overall, thirteen numbers in the Collatz map for 15 back to 1 are
composite,[81] where the largest prime in the trajectory of 120 back to {4 ➙ 2 ➙ 1 ➙ 4 ➙ ...} is the sixteenth prime number,
53.[24]
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.[85] It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).[86]
Five is
conjectured to be the only odd
untouchable number, and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.[87] Meanwhile:
Every odd number greater than is the sum of at most five prime numbers,[88] and
Every odd number greater than is conjectured to be expressible as the sum of three prime numbers.
Helfgott has provided a proof of this[89] (also known as the
odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes
peer-review.
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 .[95]: pp.1, 2 Aside from , the five classes of Ramsey permutations are the classes of:[95]: p.4
The
chromatic number of the
plane is at least five, depending on the choice of
set-theoretical 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.[104][105] 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 four-coloring
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.
Space-fillingconvex polyhedra with regular faces: the triangular prism,
hexagonal prism, cube, truncated octahedron, and
gyrobifastigium.[112] 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 solids, respectively, that can tessellate space with their own copies.
The
grand antiprism, which is the only known
non-Wythoffian construction of a uniform polychoron, is made of twenty pentagonal antiprisms and three hundred tetrahedra, with a total of one hundred vertices and five hundred edges.[118]
Overall, the fourth dimension contains five fundamental
Weyl groups that form a finite number of
uniform polychora based on only twenty-five uniform polyhedra:
,
,
,
, and
, accompanied by a fifth or sixth general group of unique
4-prisms of Platonic and Archimedean solids. There are also a total of five
Coxeter groups that generate non-prismatic
Euclidean honeycombs in 4-space, alongside five
compact hyperbolic Coxeter groups that generate five regular
compact hyperbolic honeycombs with finite
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
rank 5, with
paracompact hyperbolic solutions existing through rank 10. Likewise, analogues of four-dimensional 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
4-polytopes that have and symmetry. There are also five regular
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.[121] Only two regular projective polytopes exist in each higher dimensional space.
There are five complex
exceptional Lie algebras:
,
,
,
, and
. The
smallest of these, of
real dimension 28, can be represented in five-dimensional complex space and
projected as a
ball rolling on top of another ball, whose
motion is described in two-dimensional space.[126] 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
600-cell, whose Euclidean
norms define a quadratic form on a lattice structure
isomorphic to the optimal configuration of spheres in eight dimensions.[127] This
sphere packing lattice structure in
8-space is held by the vertex arrangement of the
521 honeycomb, one of five Euclidean honeycombs that admit
Gosset's original definition of a
semi-regular honeycomb, which includes the three-dimensional
alternated cubic honeycomb.[128][129] The smallest simple isomorphism found inside finite simple
Lie groups is ,[130] where here represents
alternating groups and classical Chevalley groups. In particular, the smallest non-solvable group is the alternating group on five letters, which is also the smallest
simple non-
abelian group.
There are five non-supersingular 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.[134] 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.[135][136] On its own, can be represented using
standard generators that further dictate a condition where .[137][138] This condition is also held by other generators that belong to the
Tits group,[139] the only
finite simple group that is a non-strict group of Lie type that can also classify as sporadic (fifth-largest of all twenty-seven by order, too). Furthermore, over the field with five elements, holds a 133-dimensional representation where 5 acts on a
commutative yet non-
associative product as a 5-
modular analogue of the
Griess algebra♮,[140] which holds the friendly giant as its
automorphism group.
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 53 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 .
A
perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[151]
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
majortriadchord 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.
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").[152]
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.[153]
Christianity
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).[154]
There are
five elements in the universe according to
Hindu cosmology: dharti, agni, jal, vayu evam akash (earth, fire, water, air and space, respectively).
The 5 Elements of traditional Chinese Wuxing.[156]
In
East Asian tradition, there are five elements: (
water,
fire,
earth,
wood, and
metal).[157] 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.[158] 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.
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.[159] The
pentagram, or five-pointed star, bears mystic significance in various belief systems including
Baháʼí,
Christianity,
Freemasonry,
Satanism,
Taoism,
Thelema, and
Wicca. In
numerology, 5 or a series of
555, is often associated with change, evolution, love and abundance.[citation needed]
Miscellaneous
"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).[160]
^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
^
abcdefgWeisstein, Eric W.
"5". mathworld.wolfram.com. Retrieved 2020-07-30.
^Bourcereau (2015-08-19).
"28". Prime Curios!. PrimePages. Retrieved 2022-10-13. The only known number which can be expressed as the sum of the first non-negative integers (1 + 2 + 3 + 4 + 5 + 6 + 7), the first primes (2 + 3 + 5 + 7 + 11) and the first non-primes (1 + 4 + 6 + 8 + 9). There is probably no other number with this property.
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.
^Alon, Noga; Grytczuk, Jaroslaw; Hałuszczak, Mariusz; Riordan, Oliver (2002).
"Nonrepetitive colorings of graphs"(PDF). Random Structures & Algorithms. 2 (3–4): 337.
doi:
10.1002/rsa.10057.
MR1945373.
S2CID5724512. A coloring of the set of edges of a graph G is called non-repetitive if the sequence of colors on any path in G is non-repetitive...In Fig. 1 we show a non-repetitive 5-coloring of the edges of P... Since, as can easily be checked, 4 colors do not suffice for this task, we have π(P) = 5.
^Wills, J. M. (1987). "The combinatorially regular polyhedra of index 2". Aequationes Mathematicae. 34 (2–3): 206–220.
doi:
10.1007/BF01830672.
S2CID121281276.
"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 triangles." Appendix II: Uniform Polyhedra.
^Kisia, S. M. (2010),
Vertebrates: Structures and Functions, Biological Systems in Vertebrates, CRC Press, p. 106,
ISBN978-1-4398-4052-8, 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.
^Cinalli, G.; Maixner, W. J.; Sainte-Rose, C. (2012-12-06).
Pediatric Hydrocephalus. Springer Science & Business Media. p. 19.
ISBN978-88-470-2121-1. The five appendages of the starfish are thought to be homologous to five human buds
^Veith (Jr.), Gene Edward; Wilson, Douglas (2009).
Omnibus IV: The Ancient World. Veritas Press. p. 52.
ISBN978-1-932168-86-0. The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter)