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Array ( [0] => {{Short description|Integer number 5}} [1] => {{Hatnote|This article is about the number. For the years, see [[5 BC]] and [[AD 5]]. For other uses, see [[5 (disambiguation)]], [[Number Five (disambiguation)]], and [[The Five (disambiguation)]].}} [2] => {{infobox number [3] => |number=5 [4] => |numeral=[[quinary]] [5] => |prime=3rd [6] => |divisor=1, 5 [7] => |roman =V, v [8] => |greek prefix=[[Wiktionary:penta-|penta-]]/[[Wiktionary:pent-|pent-]] [9] => |latin prefix=[[Wiktionary:quinque-|quinque-]]/[[Wiktionary:quinqu-|quinqu-]]/[[Wiktionary:quint-|quint-]] [10] => |lang1=[[Greek numeral|Greek]] [11] => |lang1 symbol=ε (or Ε) [12] => |lang2=[[Eastern Arabic numerals|Arabic]], [[Central Kurdish|Kurdish]] [13] => |lang2 symbol={{resize|150%|٥}} [14] => |lang3=[[Persian language|Persian]], [[Sindhi language|Sindhi]], [[Urdu numerals|Urdu]] [15] => |lang3 symbol={{resize|150%|۵}} [16] => |lang4=[[Ge'ez script|Ge'ez]] [17] => |lang4 symbol=፭ [18] => |lang5=[[Bengali language|Bengali]] [19] => |lang5 symbol={{resize|150%|৫}} [20] => |lang6=[[Kannada language|Kannada]] [21] => |lang6 symbol={{resize|150%|೫}} [22] => |lang7=[[Punjabi language|Punjabi]] [23] => |lang7 symbol={{resize|150%|੫}} [24] => |lang8=[[Chinese numeral]] [25] => |lang8 symbol=五 [26] => |lang9=[[Armenian numerals|Armenian]]|lang9 symbol=Ե|lang10=[[Devanāgarī]] [27] => |lang10 symbol={{resize|150%|५}} [28] => |lang11=[[Hebrew language|Hebrew]] [29] => |lang11 symbol={{resize|150%|ה}} [30] => |lang12=[[Khmer numerals|Khmer]] [31] => |lang12 symbol=៥ [32] => |lang13=[[Indian numerals#Telugu numerals and their names|Telugu]] [33] => |lang13 symbol={{resize|150%|౫}} [34] => |lang14=[[Indian numerals#Malayalam numerals and their names|Malayalam]] [35] => |lang14 symbol={{resize|150%|൫}} [36] => |lang15=[[Indian numerals#Tamil numerals and their names|Tamil]] [37] => |lang15 symbol={{resize|150%|௫}} [38] => |lang16=[[Thai numerals|Thai]] [39] => |lang16 symbol=๕|lang17=[[Babylonian cuneiform numerals|Babylonian numeral]]|lang17 symbol=𒐙|lang18=[[Egyptian numerals|Egyptian hieroglyph]], [[counting rods|Chinese counting rod]]|lang18 symbol={{!}}{{!}}{{!}}{{!}}{{!}}|lang19=[[Maya numerals]]|lang19 symbol=𝋥|lang20=[[Morse code]]|lang20 symbol={{resize|150%|.....}}|cardinal=five|ordinal=5th [40] => (fifth)}} [41] => [42] => '''5''' ('''five''') is a [[number]], [[numeral (linguistics)|numeral]] and [[numerical digit|digit]]. It is the [[natural number]], and [[cardinal number]], following [[4]] and preceding [[6]], and is a [[prime number]]. It has garnered attention throughout history in part because distal [[Limb (anatomy)|extremities]] in humans typically contain five [[Digit (anatomy)|digits]]. [43] => [44] => == Evolution of the Arabic digit == [45] => [[File:Evolution5glyph.png|x45px|left]] [46] => [47] => The evolution of the modern Western digit for the numeral 5 cannot be traced back to the [[Brahmi numerals|Indian system]], as opposed to digits 1 to 4. The [[Kushan Empire|Kushana]] and [[Gupta Empire|Gupta]] empires in what is now [[India]] had among themselves several forms that bear no resemblance to the modern digit. The [[Devanagari|Nagari]] and [[Punjabi language|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.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. [48] => [49] => [[File:Seven-segment 5.svg|30px|left]] [50] => [51] => While the shape of the character for the digit 5 has an [[Ascender (typography)|ascender]] in most modern [[typeface]]s, in typefaces with [[text figures]] the glyph usually has a [[descender]], as, for example, in [[File:Text figures 256.svg|45px]]. [52] => [53] => 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. [54] => [55] => == Mathematics == [56] => [[File:Pythagoras' Special Triples.svg|right|110px|thumb|The first [[Pythagorean triple]], with a [[hypotenuse]] of

5

]] [57] => [58] => '''Five''' is the third-smallest [[prime number]], and the second [[super-prime]].{{Cite web|last=Weisstein|first=Eric W.|title=5|url=https://mathworld.wolfram.com/5.html|access-date=2020-07-30|website=mathworld.wolfram.com|language=en}} It is the first [[safe prime]],{{Cite OEIS |A005385 |Safe primes p: (p-1)/2 is also prime |access-date=2023-02-14 }} the first [[good prime]],{{Cite OEIS |A007540 |Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2). |access-date=2023-09-06 }} the first [[balanced prime]],{{Cite OEIS |A006562 |Balanced primes (of order one): primes which are the average of the previous prime and the following prime. |access-date=2023-02-14 }} and the first of three known [[Wilson prime]]s.{{Cite OEIS|1=A028388 |2=Good primes|access-date=2016-06-01}} Five is the second [[Fermat prime]], the second [[Proth prime]],{{Cite OEIS |A080076 |Proth primes |access-date=2023-06-21 }} and the third [[Mersenne prime]] exponent,{{Cite web|last=Weisstein|first=Eric W.|title=Mersenne Prime|url=https://mathworld.wolfram.com/MersennePrime.html|access-date=2020-07-30|website=mathworld.wolfram.com|language=en}} as well as the third [[Catalan number]]{{Cite web|last=Weisstein|first=Eric W.|title=Catalan Number|url=https://mathworld.wolfram.com/CatalanNumber.html|access-date=2020-07-30|website=mathworld.wolfram.com|language=en}} and the third [[Sophie Germain prime]]. Notably, 5 is equal to the sum of the ''only'' consecutive primes [[2]] + [[3]] and it is the only number that is part of more than one pair of [[twin prime]]s, ([[3]], 5) and (5, [[7]]).{{Cite OEIS |A001359 |Lesser of twin primes. |access-date=2023-02-14 }}{{Cite OEIS |A006512 |Greater of twin primes. |access-date=2023-02-14 }} It also forms the first pair of [[sexy prime]]s with [[11 (number)|11]],{{Cite OEIS |A023201 |Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.) |access-date=2023-01-14 }} which is the fifth prime number and [[Heegner number]],{{Cite OEIS |A003173 |Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1). |access-date=2023-06-20 }} as well as the first [[repunit prime]] in [[Base ten|decimal]]; a base in-which five is also the first non-trivial 1-[[automorphic number]].{{Cite OEIS |A003226 |Automorphic numbers: m^2 ends with m. |access-date=2023-05-26 }} Five is the third [[factorial prime]],{{Cite OEIS |A088054 |Factorial primes: primes which are within 1 of a factorial number. |access-date=2023-02-14 }} and an [[alternating factorial]].{{Cite web|last=Weisstein|first=Eric W.|title=Twin Primes|url=https://mathworld.wolfram.com/TwinPrimes.html|access-date=2020-07-30|website=mathworld.wolfram.com|language=en}} It is also an [[Eisenstein prime]] (like 11) with no [[Imaginary number|imaginary]] part and [[Real number|real]] part of the form

3p - 1

. In particular, five is the first [[congruent number]], since it is the length of the [[hypotenuse]] of the smallest [[Pythagorean triple|integer-sided right triangle]].{{Cite OEIS|1=A003273 |2=Congruent numbers|access-date=2016-06-01}} [59] => [60] => === Number theory === [61] => 5 is the fifth [[Fibonacci number]], being 2 plus 3, 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 (number)|13]]), (2, 5, [[29 (number)|29]]), (5, 13, [[194 (number)|194]]), (5, 29, 433), ... ({{OEIS2C|id=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 number]]s.{{Cite web|last=Weisstein|first=Eric W.|title=Perrin Sequence|url=https://mathworld.wolfram.com/PerrinSequence.html|access-date=2020-07-30|website=mathworld.wolfram.com|language=en}} [62] => [63] => 5 is the second [[Fermat number|Fermat prime]] of the form

2^{2^{n}} + 1

, and more generally the second [[OEIS:A014566|Sierpiński number of the first kind]],

n^n + 1

.{{Cite web|last=Weisstein|first=Eric W.|title=Sierpiński Number of the First Kind|url=https://mathworld.wolfram.com/SierpinskiNumberoftheFirstKind.html|access-date=2020-07-30|website=mathworld.wolfram.com|language=en}} There are a total of five known Fermat primes, which also include [[3]], [[17 (number)|17]], [[257 (number)|257]], and [[65537 (number)|65537]].{{Cite OEIS |A019434 |Fermat primes |access-date=2022-07-21 }} The sum of the first three Fermat primes, 3, '''5''' and 17, yields 25 or 5², while 257 is the 55th prime number. Combinations from these five Fermat primes generate [[31 (number)|thirty-one]] polygons with an [[Parity (mathematics)|odd]] number of [[Edge (geometry)|sides]] that can be [[Constructible polygon|constructed purely with a compass and straight-edge]], which includes the five-sided [[regular pentagon]].{{Cite OEIS |A004729 |... the 31 regular polygons with an odd number of sides constructible with ruler and compass |access-date=2023-05-26 }}{{Cite book |last1=Conway |first1=John H. |author1-link= John Horton Conway |last2=Guy |first2=Richard K. |author-link2=Richard K. Guy |title=The Book of Numbers |url=https://link.springer.com/book/10.1007/978-1-4612-4072-3 |year=1996 |publisher=Copernicus ([[Springer Publishing|Springer]]) |location=New York, NY |pages=ix; 1–310 |isbn=978-1-4612-8488-8 |doi=10.1007/978-1-4612-4072-3 |oclc=32854557 |s2cid=115239655 }}{{rp|pp.137–142}} Apropos, thirty-one is also equal to the sum of the maximum number of [[area]]s inside a circle that are formed from the [[Edge (geometry)|sides]] and [[diagonal]]s of the first five

n

-sided [[polygon]]s, which is equal to the maximum number of areas formed by a six-sided polygon; per [[Moser's circle problem]].{{Cite OEIS |A000127 |Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes. |access-date=2022-10-31 }}{{rp|pp.76–78}} [64] => [65] => 5 is also the third [[Mersenne prime]] exponent of the form

2^{n} - 1

, which yields

31

, the eleventh prime number and fifth [[super-prime]].{{Cite OEIS |A000040 |The prime numbers. |access-date=2023-11-08 }}{{Cite OEIS |A000668 |Mersenne primes (primes of the form 2^n - 1). |access-date=2023-07-03 }} This is the [[List of prime numbers#The first 1000 prime numbers|prime index]] of the third [[Mersenne prime]] and second [[double Mersenne number|double Mersenne prime]] [[127 (number)|127]],{{Cite OEIS |A103901 |Mersenne primes p such that M(p) equal to 2^p - 1 is also a (Mersenne) prime. |access-date=2023-07-03 }} as well as the third double Mersenne prime exponent for the number [[2,147,483,647]], which is the largest value that a [[Signed number representations|signed]] [[32-bit]] [[Integer (computer science)|integer field]] can hold. There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime

M_{M_{61}}

= 2^{23058...93951} − 1 too large to compute with current computers. In a related sequence, the first five terms in the sequence of [[Double Mersenne number#Catalan–Mersenne numbers|Catalan–Mersenne numbers]]

M_{c_{n}}

are the only known prime terms, with a sixth possible candidate in the order of 10^{10^37.7094}. These prime sequences are conjectured to be prime up to a certain limit. [66] => [67] => There are a total of five known [[unitary perfect number]]s, which are numbers that are the sums of their positive [[Proper divisor|proper]] [[unitary divisor]]s.{{cite book|author=Richard K. Guy|author-link=Richard K. Guy|title=Unsolved Problems in Number Theory|publisher=[[Springer-Verlag]]|year=2004|isbn=0-387-20860-7 | pages=84–86}}{{Cite OEIS |A002827 |Unitary perfect numbers |access-date=2023-01-10 }} 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 equation#Triangular Mersenne numbers|Ramanujan–Nagell numbers]] that are both triangular numbers and ''Mersenne numbers'' of the general form.{{Cite OEIS |A076046 |Ramanujan-Nagell numbers: the triangular numbers...which are also of the form 2^b - 1 |access-date=2023-01-10 }}{{Cite OEIS |A000225 |... (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.) |access-date=2023-01-13 }} The sums of the first five non-primes greater than zero {{math|[[1]] + [[4]] + [[6]] + [[8]] + [[9]]}} and the first five prime numbers {{math|[[2]] + [[3]] + '''5''' + [[7]] + [[11 (number)|11]]}} both equal [[28 (number)|28]]; the seventh [[triangular number]] and like 6 a [[perfect number]], which also includes [[496 (number)|496]], the thirty-first triangular number and perfect number of the form

2^{p-1}

(

2^{p}-1

) with a

p

5

, by the [[Euclid–Euler theorem]].{{Cite web|url=https://primes.utm.edu/curios/page.php/28.html |title=28 |date=2015-08-19 |website=Prime Curios! |publisher=PrimePages |author=Bourcereau |access-date=2022-10-13 |quote=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.}}{{Cite OEIS |A000396 |Perfect numbers k: k is equal to the sum of the proper divisors of k. |access-date=2022-10-13 }}{{Cite OEIS |A000217 |Triangular numbers. |access-date=2022-10-13 }} Within the larger family of [[Harmonic divisor number|Ore numbers]], [[140 (number)|140]] and 496, respectively the fourth and sixth [[Sequence|indexed]] members, both contain a set of [[divisor]]s that produce integer [[harmonic mean]]s equal to 5.{{Cite OEIS |A001599 |Harmonic or Ore numbers: numbers n such that the harmonic mean of the divisors of n is an integer. |access-date=2022-12-26 }}{{Cite OEIS |A001600 |Harmonic means of divisors of harmonic numbers. |access-date=2022-12-26 }} The fifth Mersenne prime, [[8191 (number)|8191]], splits into 4095 and [[4096 (number)|4096]], with the latter being the fifth [[Superperfect number|superperfect]] number{{Cite OEIS |A019279 |Superperfect numbers: numbers k such that sigma(sigma(k)) equals 2*k where sigma is the sum-of-divisors function. |access-date=2023-07-26 }} and the sixth power of four, 4⁶. [68] => [69] => ==== Figurate numbers ==== [70] => [71] => In [[figurate number]]s, 5 is a [[pentagonal number]], with the [[sequence (mathematics)|sequence]] of pentagonal numbers starting: 1, '''5''', 12, 22, 35, '''...'''{{Cite OEIS |A000326 |Pentagonal numbers. |access-date=2022-11-08 }} [72] => {{Bullet list [73] => |5 is a [[centered tetrahedral number]]: 1, '''5''', 15, 35, 69, '''...'''{{Cite OEIS |A005894 |Centered tetrahedral numbers. |access-date=2022-11-08 }} Every centered tetrahedral number with an index of 2, 3 or 4 [[Modulo operation|modulo]] 5 is divisible by 5. [74] => |5 is a [[square pyramidal number]]: 1, '''5''', 14, 30, 55, '''...'''{{Cite OEIS |A000330 |Square pyramidal numbers. |access-date=2022-11-08 }} The first four members add to [[50 (number)|50]] while the fifth [[Sequence (mathematics)|indexed]] member in the sequence is [[55 (number)|55]]. [75] => |5 is a [[centered square number]]: 1, '''5''', 13, 25, 41, '''...'''{{Cite OEIS |A001844 |Centered square numbers...Sum of two squares. |access-date=2022-11-08 }} The fifth [[square number]] or 5² is [[25 (number)|25]], which features in the proportions of the two smallest (3, 4, '''5''') and ('''5''', 12, 13) ''primitive'' [[Pythagorean triple]]s.{{Cite OEIS |A103606 |Primitive Pythagorean triples in nondecreasing order of perimeter, with each triple in increasing order, and if perimeters coincide then increasing order of the even members. |access-date=2023-05-26 }} [76] => }} [77] => [78] => The [[factorial]] of five

5! = 120

is [[Multiply perfect number|multiply perfect]] like 28 and 496.{{Cite OEIS |A007691 |Multiply-perfect numbers: n divides sigma(n). |access-date=2023-06-28 }} It is the sum of the first [[15 (number)|fifteen]] non-zero positive [[integer]]s and 15th [[triangular number]], which in-turn is the sum of the first '''five''' non-zero positive integers and 5th triangular number. Furthermore,

120 + 5 = 125 = 5^{3}

, where [[125 (number)|125]] is the second number to have an [[aliquot sum]] of 31 (after the fifth [[Exponentiation|power]] of [[2|two]], 32).{{Cite OEIS |A001065 |Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n. |access-date=2023-08-11 }} On its own, 31 is the first prime [[centered pentagonal number]],{{Cite OEIS |A005891 |Centered pentagonal numbers |access-date=2023-06-21 }} and the fifth [[centered triangular number]].{{Cite OEIS |A005448 |Centered triangular numbers |access-date=2023-06-21 }} Collectively, five and thirty-one generate a sum of [[36 (number)|36]] (the [[Square number|square]] of [[6]]) and a [[subtraction|difference]] of [[26 (number)|26]], which is the only number to lie between a square

a^{2}

and a [[Cube (algebra)|cube]]

b^{3}

(respectively, 25 and [[27 (number)|27]]).{{Cite web |last=Conrad |first=Keith E. |title=Example of Mordell's Equation |url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/mordelleqn1.pdf |type=Professor Notes |publisher=[[University of Connecticut]] (Homepage) |page=10 |s2cid=5216897 }} The fifth pentagonal and [[tetrahedral number]] is [[35 (number)|35]], which is equal to the sum of the first five triangular numbers: 1, 3, 6, 10, 15.{{Cite OEIS |A000217 |Triangular numbers. |access-date=2022-11-08 }} In general, the sum of ''n'' consecutive triangular numbers is the ''nth'' tetrahedral number. In the sequence of [[pentatope number]]s 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 (number)|70]], 126, 210, 330, 495, ...{{Cite OEIS |A000332 |Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}... |access-date=2023-06-14 }} The first five [[Sequence|members]] in this sequence add to [[126 (number)|126]], which is also the sixth [[pentagonal pyramidal number]]{{Cite OEIS |A002411 |Pentagonal pyramidal numbers |access-date=2023-06-28 }} as well as the fifth

\mathcal{S}

-perfect [[Granville number]].{{Cite OEIS |A118372 |S-perfect numbers. |access-date=2023-06-28 }} This is the third Granville number not to be ''perfect'', and the only known such number with three distinct prime factors.{{Cite book |last=de Koninck |first=Jean-Marie |author-link=Jean-Marie De Koninck |translator-first = J. M. |translator-last=de Koninck |title=Those Fascinating Numbers |url=https://archive.org/details/thosefascinating0000koni/page/40/mode/2up |url-access=registration |publisher=[[American Mathematical Society]] |location= Providence, RI |year=2008 |page=40 |isbn=978-0-8218-4807-4 |oclc=317778112 |mr=2532459 }} [79] => [80] => '''[[55 (number)|55]]''' is the fifteenth discrete [[biprime]],{{Cite OEIS |A006881 |Squarefree semiprimes: Numbers that are the product of two distinct primes. |access-date=2023-09-06 }} equal to the product between 5 and the fifth prime and third super-prime [[11 (number)|11]]. These two numbers also form the second pair (5, 11) of [[Brocard's problem|Brown numbers]]

(n,m)

such that

n!+1 = m^2

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 (number)|71]]).{{Cite OEIS |A216071 |Brocard's problem: positive integers m such that m^2 equal to n! + 1 for some n. |access-date=2023-09-09 }}{{Cite OEIS |A085692 |Brocard's problem: squares which can be written as n!+1 for some n. |access-date=2023-09-09 }} [81] => Fifty-five is also the tenth [[Fibonacci number]],{{Cite OEIS |A000045 |Fibonacci numbers: F(n) is F(n-1) + F(n-2) with F(0) equal to 0 and F(1) equal to 1. |access-date=2023-09-06 }} whose [[digit sum]] is also [[10]], in its [[decimal]] representation. It is the tenth triangular number and the fourth that is [[Doubly triangular number|doubly triangular]],{{Cite OEIS |A002817 |Doubly triangular numbers: a(n) equal to n*(n+1)*(n^2+n+2)/8. |access-date=2023-09-06 }} the fifth [[heptagonal number]]{{Cite OEIS |A000566 |Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2. |access-date=2023-09-06 }} and fourth [[centered nonagonal number]],{{Cite OEIS |A060544 |Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1) equal to 1 |access-date=2023-09-06 }} and as listed above, the fifth square pyramidal number. The sequence of triangular

n

that are [[exponentiation|powers]] of 10 is: 55, '''[[5050 (number)|5050]]''', [[100,000 (number)#500,000 to 599,999|500500]], ...{{Cite OEIS |A037156 |a(n) equal to 10^n*(10^n+1)/2. |access-date=2023-09-06 }}
a(0) = 1 = 1 * 1 = 1
a(1) = 1 + 2 + ...... + 10 = 11 * 5 = 55
a(2) = 1 + 2 + .... + 100 = 101 * 50 = 5050
a(3) = 1 + 2 + .. + 1000 = 1001 * 500 = 500500
... 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 (number)|45]],{{Cite OEIS |A006886 |Kaprekar numbers... |access-date=2023-09-07 }} with forty-five itself the ninth triangular number where 5 lies midway between 1 and 9 in the sequence of [[natural number]]s. 45 is also conjectured by [[Ramsey's theorem#Ramsey numbers|Ramsey number]]

R(5, 5)

,{{Cite OEIS |A120414 |Conjectured Ramsey number R(n,n). |access-date=2023-09-07 }}{{Cite OEIS |A212954 |Triangle read by rows: two color Ramsey numbers |access-date=2023-09-07 }} and is a [[Schröder–Hipparchus number]]; the next and fifth such number is [[197 (number)|197]], the forty-fifth prime number that represents the number of ways of dissecting a [[heptagon]] into smaller polygons by inserting [[diagonal]]s.{{Cite OEIS |A001003 |Schroeder's second problem; ... also called super-Catalan numbers or little Schroeder numbers. |access-date=2023-09-07 }} A five-sided [[Convex polygon|convex]] [[pentagon]], on the other hand, has eleven ways of being subdivided in such manner.{{efn|1=[[File:Pentagon subdivisions.svg|163px]] }} [82] => [83] => ==== Magic figures ==== [84] => [85] => [[File:Magic Square Lo Shu.svg|110px|right|thumb|The smallest non-trivial [[magic square]]]] [86] => [87] => 5 is the value of the central [[Magic square#Properties of magic squares|cell]] of the first non-trivial [[magic square|normal magic square]], called the [[Luoshu Square|''Luoshu'' square]]. Its

3 \times 3

array has a [[magic constant]]

\mathrm {M}

15

, where the sums of its rows, columns, and diagonals are all equal to fifteen.{{Cite web |url=https://www.math.wichita.edu/~richardson/mathematics/magic%20squares/order3magicsquare.html |title=Magic Squares of Order 3 |author=William H. Richardson |website=Wichita State University Dept. of Mathematics |access-date=2022-07-14 }} On the other hand, a normal

5 \times 5

magic square{{efn|1=

[88] => \begin{bmatrix}
    [89] =>  17 & 24 & 1 & 8 & 15 \\
    [90] =>  23 & 5 & 7 & 14 & 16 \\
    [91] =>  4 & 6 & 13 & 20 & 22 \\
    [92] =>  10 & 12 & 19 & 21 & 3 \\
    [93] =>  11 & 18 & 25 & 2 & 9
    [94] => \end{bmatrix}

}} has a magic constant

\mathrm {M}

65 = 13 \times 5

, where '''5''' and [[13 (number)|13]] are the first two [[Wilson prime]]s. The fifth number to return

0

for the [[Mertens function]] is [[65 (number)|65]],{{Cite OEIS |A028442 |Numbers k such that Mertens's function M(k) (A002321) is zero. |access-date=2023-09-06 }} with

M(x)

counting the number of [[square-free integer]]s up to

x

with an ''even'' number of [[prime factor]]s, minus the count of numbers with an ''odd'' number of prime factors. 65 is the nineteenth biprime with distinct prime factors, with an aliquot sum of [[19 (number)|19]] as well and equivalent to {{math|1=1⁵ + 2⁴ + 3³ + 4² + 5¹ }}.{{Cite OEIS |A003101 |a(n) as Sum_{k equal to 1..n} (n - k + 1)^k. |access-date=2023-09-06 }} It is also the magic constant of the

n-

[[Eight queens puzzle|Queens Problem]] for

n = 5

,{{Cite OEIS|A006003 |a(n) equal to n*(n^2 + 1)/2. |access-date=2023-09-06 }} the fifth [[octagonal number]],{{Cite OEIS |A000567 |Octagonal numbers: n*(3*n-2). Also called star numbers. |access-date=2023-09-06 }} and the [[Stirling number of the second kind]]

S(6,4)

that represents sixty-five ways of dividing a [[Set theory|set]] of six objects into four non-empty [[subset]]s.{{Cite web|url=https://oeis.org/A008277|title=Sloane's A008277 :Triangle of Stirling numbers of the second kind|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2021-12-24}} 13 and 5 are also the fourth and third [[Markov number]]s, respectively, where the sixth member in this sequence ([[34 (number)|34]]) is the magic constant of a normal [[Magic star|magic octagram]] and

4 \times 4

magic square.{{Cite OEIS |A006003 |a(n) equal to n*(n^2 + 1)/2. |access-date=2023-09-11 }} In between these three Markov numbers is the tenth prime number [[29 (number)|29]] that represents the number of [[Polycube#Enumerating polycubes|pentacubes]] when [[Chirality (mathematics)|reflections]] are considered distinct; this number is also the fifth [[Lucas number#Lucas primes|Lucas prime]] after 11 and 7 (where the first prime that is not a Lucas prime is 5, followed by 13).{{Cite OEIS |A000162 |Number of 3-dimensional polyominoes (or polycubes) with n cells. |access-date=2023-09-11 }} A magic constant of '''[[505 (number)|505]]''' is generated by a

10 \times 10

normal magic square, where [[10]] is the fifth [[Composite number|composite]].{{Cite OEIS |A002808 |The composite numbers: numbers n of the form x*y for x > 1 and y > 1. |access-date=2023-09-25 }} [95] => [96] => 5 is also the value of the central cell the only non-trivial [[magic hexagon|normal magic hexagon]] made of nineteen cells.{{Cite magazine |url=http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/magic-hexagon-trigg |author=Trigg, C. W. |title= A Unique Magic Hexagon |magazine= Recreational Mathematics Magazine |date=February 1964 |access-date=2022-07-14 }}{{efn|1=[[File:MagicHexagon-Order3-a.svg|163px]] }} Where the sum between the magic constants of this order-3 normal magic hexagon ([[38 (number)|38]]) and the order-5 normal magic square (65) is [[103 (number)|103]] — the [[Sequence|prime index]] of the third Wilson prime [[563 (number)|563]] equal to the sum of all three pairs of Brown numbers — their difference is 27, itself the prime index of 103. In base-ten, 15 and [[27 (number)|27]] are the only two-digit numbers that are equal to the sum between their [[Numerical digit|digits]] (inclusive, i.e. 2 + 3 + ... + 7 = 27), with these two numbers consecutive [[perfect totient number]]s after [[3]] and [[9]].{{Cite OEIS |A082897 |Perfect totient numbers. |access-date=2023-09-10 }} 103 is the fifth [[Regular prime#Irregular primes|irregular prime]]{{Cite OEIS |A000928 |Irregular primes: primes p such that at least one of the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) is divisible by p. |access-date=2023-09-07 }} that divides the [[numerator]] (236364091) of the twenty-fourth [[Bernoulli number]]

B_{24}

, and as such it is part of the eighth [[Regular prime#irregular pair|irregular pair]] (103, 24).{{Cite OEIS |A189683 |Irregular pairs (p,2k) ordered by increasing k. |access-date=2023-09-07 }} In a two-dimensional array, the number of [[Plane partition|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,{{Cite OEIS |A000219 |Number of planar partitions (or plane partitions) of n. |access-date=2023-09-10 }} a value equal to the [[Divisor function|sum-of-divisors]] of the ninth [[arithmetic number]] 15{{Cite OEIS |A000203 |a(n) is sigma(n), the sum of the divisors of n. Also called sigma_1(n). |access-date=2023-09-14 }} whose divisors also produce an integer [[arithmetic mean]] of [[6]]{{Cite OEIS |A102187 |Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer). |access-date=2023-09-14 }} (alongside an [[aliquot sum]] of 9). The smallest value that the magic constant of a five-pointed [[Magic star|magic pentagram]] can have using distinct integers is 24.{{Cite book |last=Gardner |first=Martin |author-link=Martin Gardner|title=Mathematical Carnival |url=https://bookstore.ams.org/view?ProductCode=GARDNER/6 |series=Mathematical Games |publisher=[[Mathematical Association of America]] |location=Washington, D.C. |edition=5th |year=1989 |pages=56–58 |isbn=978-0-88385-448-8 |oclc=20003033 |zbl=0684.00001 }}{{efn|1=[[File:Magic Pentagram.png|158px]] }} [97] => [98] => ==== Collatz conjecture ==== [99] => [100] => In the [[Collatz conjecture|Collatz]] '''{{math|3''x'' + 1}}''' [[Collatz conjecture|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 (number)|32]] since 16 ''must'' be part of such path (see {{efn|1=[[File:Collatz-graph-50-no27.svg|180px]] }} for a map of [[Orbit (dynamics)|orbits]] for small odd numbers).{{Cite web |editor-last=Sloane |editor-first= N. J. A. |editor-link=Neil Sloane |url=http://oeis.org/wiki/3x%2B1_problem |title=3x+1 problem |website=The [[On-Line Encyclopedia of Integer Sequences]] |publisher=The OEIS Foundation |access-date=2023-01-24 }}{{Cite OEIS |A006577 |Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached. |access-date=2023-01-24 }} [101] => :"Table of n, a(n) for n = 1..10000" [102] => [103] => 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 (number)|16]] is the smallest number with exactly five divisors,{{Cite OEIS |A005179 |Smallest number with exactly n divisors. |access-date=2023-11-06 }} and one of only two numbers to have an aliquot sum of [[15 (number)|15]], the other being [[33 (number)|33]]. Otherwise, the trajectory of 15 requires seventeen steps to reach 1, where its ''reduced'' Collatz trajectory is equal to five when counting the steps {23, 53, 5, 2, 1} that are prime, including 1.{{Cite OEIS |A286380 |a(n) is the minimum number of iterations of the reduced Collatz function R required to yield 1. The function R (A139391) is defined as R(k) equal to (3k+1)/2^r, with r as large as possible. |access-date=2023-09-18 }} Overall, thirteen numbers in the Collatz map for 15 back to 1 are [[Composite number|composite]], where the largest prime in the trajectory of 120 back to {4 ➙ 2 ➙ 1 ➙ 4 ➙ ...} is the sixteenth prime number, [[53 (number)|53]]. [104] => [105] => When generalizing the [[Collatz conjecture#Extensions to larger domains|Collatz conjecture]] to all positive or negative [[integer]]s, '''−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 {{math|3''x'' − 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.{{Cite OEIS |A003079 |One of the basic cycles in the x->3x-1 (x odd) or x/2 (x even) problem |access-date=2023-01-24 }} [106] => :{5 ➙ 14 ➙ 7 ➙ 20 ➙ 10 ➙ 5 ➙ ...}. It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).{{Cite web |editor-last=Sloane |editor-first= N. J. A. |editor-link=Neil Sloane |url=http://oeis.org/wiki/3x-1_problem |title=3x-1 problem |website=The [[On-Line Encyclopedia of Integer Sequences]] |publisher=The OEIS Foundation |access-date=2023-01-24 }} [107] => [108] => ==== Generalizations ==== [109] => {{Unsolved|mathematics|Is 5 the only odd untouchable number?}} [110] => [111] => Five is [[conjecture]]d 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.{{Cite journal|last=Pomerance|first=Carl|title=On Untouchable Numbers and Related Problems|url=https://math.dartmouth.edu/~carlp/uupaper3.pdf|journal=[[Dartmouth College]]|page=1|year=2012|s2cid=30344483}} Meanwhile: [112] => [113] => {{Bullet list [114] => |Every odd number greater than

1

is the sum of at most five prime numbers,{{Cite journal |last=Tao |first=Terence |date=March 2014 |title=Every odd number greater than 1 is the sum of at most five primes |url=https://www.ams.org/journals/mcom/2014-83-286/S0025-5718-2013-02733-0/S0025-5718-2013-02733-0.pdf |journal=Mathematics of Computation |volume=83 |number=286 |pages=997–1038 |doi=10.1090/S0025-5718-2013-02733-0 |mr=3143702 |s2cid=2618958 }} and [115] => |Every odd number greater than

5

is conjectured to be expressible as the sum of three prime numbers. [[Harald Andres Helfgott|Helfgott]] has provided a proof of this{{Cite book |editor-last=Jang |editor-first=Sun Young |last=Helfgott |first=Harald Andres |date=2014 |chapter=The ternary Goldbach problem |chapter-url=https://www.imj-prg.fr/wp-content/uploads/2020/prix/helfgott2014.pdf |title=Seoul [[International Congress of Mathematicians]] Proceedings |volume=2 |publisher=Kyung Moon SA |location=Seoul, KOR |pages=391–418 |isbn=978-89-6105-805-6 |oclc=913564239 }} (also known as the [[Goldbach's weak conjecture|odd Goldbach conjecture]]) that is already widely acknowledged by mathematicians as it still undergoes [[peer-review]]. [116] => }} [117] => [118] => As a consequence of [[Fermat's little theorem]] and [[Euler's criterion]], all squares are [[Modular arithmetic#Congruence|congruent]] to [[0]], [[1]], 4 (or [[−1]]) [[Modulo operation|modulo]] 5.{{Cite journal |first=James A. |last=Sellers |title=An unexpected congruence modulo 5 for 4-colored generalized Frobenius partitions. |journal=J. Indian Math. Soc. |volume=New Series |year=2013 |issue=Special Issue |page=99 |publisher=[[Indian Mathematical Society]] |location=Pune, IMD |bibcode=2013arXiv1302.5708S |arxiv=1302.5708 |mr=157339|zbl=1290.05015 |s2cid=116931082 }} In particular, all [[integer]]s

n \ge 34

can be expressed as the sum of five non-zero [[Square number|squares]].{{Cite book |last1= Niven |first1=Ivan |author1-link=Ivan M. Niven |last2=Zuckerman |first2= Herbert S. |last3= Montgomery |first3= Hugh L. |author3-link=Hugh Lowell Montgomery |title=An Introduction to the Theory of Numbers |publisher=[[Wiley (publisher)|John Wiley]] |location=New York, NY |edition=5th |year=1980 |pages=144, 145 |isbn=978-0-19-853171-5 }}{{Cite OEIS |A047701 |All positive numbers that are not the sum of 5 nonzero squares. |access-date=2023-09-20 }} [119] => :Only twelve integers up to [[33 (number)|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. [120] => [121] => Regarding [[Waring's problem]],

g(5) = 37

, where every [[natural number]]

n \in \mathbb {N}

is the sum of at most thirty-seven [[Fifth power (algebra)|fifth powers]].{{Cite OEIS |A002804 |(Presumed) solution to Waring's problem: g(n) equal to 2^n + floor((3/2)^n) - 2. |access-date=2023-09-20 }}{{Cite journal |first=David |last=Hilbert |author-link=David Hilbert |title=Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem) |url=https://zenodo.org/record/1428266 |language=de |journal=[[Mathematische Annalen]] |volume=67 |pages=281–300 |year=1909 |issue=3 |doi=10.1007/bf01450405 |doi-access=free |mr=1511530|s2cid = 179177986 |jfm=40.0236.03 }} [122] => [123] => There are five countably infinite [[Ramsey class]]es of [[permutation]]s, where the [[Fraïssé limit#Finitely generated substructures and age|age]] of each countable homogeneous permutation forms an individual Ramsey class

K

of [[Group object|objects]] such that, for each natural number

r

and each choice of objects

A,B \in K

, there is no object

C \in K

where in any

r

-[[Edge coloring|coloring]] of all [[subobject]]s of

C

[[isomorphism|isomorphic]] to

A

there exists a [[Graph coloring|monochromatic]] subobject isomorphic to

B

.{{Cite journal |last1=Böttcher |first1=Julia |last2=Foniok |first2=Jan |title=Ramsey Properties of Permutations |url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p2 |journal=The Electronic Journal of Combinatorics |volume=20 |issue=1 |year=2013 |page=P2 |doi=10.37236/2978 |arxiv=1103.5686v2 |s2cid=17184541 |zbl=1267.05284 }}{{rp|pp.1,2}} Aside from

\{1\}

, the five classes of Ramsey permutations are the classes of:{{rp|p.4}} [124] => {{Bullet list [125] => |[[Permutation group#Neutral element and inverses|Identity permutations]], and reversals [126] => |[[Sequence#Formal definition and basic properties|Increasing sequences]] of decreasing sequences, and [[Sequence#Formal definition and basic properties|decreasing sequences]] of increasing sequences [127] => |'''All''' permutations [128] => }} [129] => [130] => In general, the [[Fraïssé limit]] of a class

K

of finite [[Structure (mathematical logic)|relational structure]] is the age of a countable homogeneous relational structure

U

[[if and only if]] five conditions hold for

K

: it is [[Isomorphism-closed subcategory|closed under isomorphism]], it has only countably many [[isomorphism class]]es, it is [[Hereditary property|hereditary]], it is [[Joint embedding property|joint-embedded]], and it holds the [[amalgamation property]].{{rp|p.3}} [131] => [132] => [[Polynomial]] equations of degree {{num|4}} and below can be solved with radicals, while [[quintic equation]]s of degree 5 and higher cannot generally be so solved (see, [[Abel–Ruffini theorem]]). This is related to the fact that the [[symmetric group]]

\mathrm{S}_{n}

is a [[solvable group]] for

n

⩽

4

, and not for

n

⩾

5

. [133] => [134] => In the general [[Number#Main classification|classification of number systems]], the [[real numbers]]

\mathbb{R}

and its three subsequent [[Cayley–Dickson construction|Cayley–Dickson constructions]] of [[algebra]]s over the field of the real numbers (i.e. the [[complex numbers]]

\mathbb C

, the [[quaternion]]s

\mathbb H

, and the [[octonion]]s

\mathbb O

) are [[normed division algebra]]s that hold up to five different principal algebraic properties of interest: whether the algebras are [[Ordered field|ordered]], and whether they hold [[Commutative property|commutative]], [[Associative property|associative]], [[Alternative algebra|alternative]], and [[Power associativity|power-associative]] multiplicative properties.{{Cite book |last1=Kantor |first1=I. L. |last2=Solodownikow |first2=A. S. |translator-last1=Shenitzer. |translator-first1=A. |title= Hypercomplex Numbers: An Elementary Introduction to Algebras |publisher=[[Springer Science+Business Media|Springer-Verlag]] |location=New York, NY | year=1989 |pages=109–110 |isbn=978-1-4612-8191-7 |oclc=19515061 |s2cid=60314285 }} Whereas the real numbers contain all five properties, the octonions are only alternative and power-associative. In comparison, the [[sedenion]]s

\mathbb S

, which represent a fifth algebra in this series, is not a [[composition algebra]] unlike

\mathbb H

and

\mathbb O

, is only power-associative, and is the first algebra to contain non-trivial [[zero divisor]]s as with all further algebras over larger fields.{{Cite journal |last1=Imaeda |first1=K. |last2=Imaeda |first2=M. |title=Sedenions: algebra and analysis |journal=Applied Mathematics and Computation |publisher=[[Elsevier]] |location=Amsterdam, Netherlands |volume=115 |issue=2 |year=2000 |pages=77–88 |doi=10.1016/S0096-3003(99)00140-X |mr=1786945 |s2cid=32296814 |zbl=1032.17003 }} Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16. [135] => [136] => === Geometry === [137] => [[File:Five Pointed Star Lined.svg|left|105px]] [138] => [139] => A [[pentagram]], or five-pointed [[Polygram (geometry)|polygram]], is the first proper [[star polygon]] constructed from the diagonals of a [[regular pentagon]] as [[Star polygon#Regular star polygon|self-intersecting edges]] that are proportioned in [[golden ratio]], [[Golden ratio|

\varphi

]]. Its internal geometry appears prominently in [[Penrose tilings]], and is a [[Facet (geometry)|facet]] inside [[Kepler–Poinsot polyhedron|Kepler–Poinsot star polyhedra]] and [[Regular 4-polytope#Regular star (Schläfli–Hess) 4-polytopes|Schläfli–Hess star polychora]], represented by its [[Schläfli symbol]] {{math|1={5/2} }}. A similar figure to the pentagram is a [[Five-pointed star|five-pointed]] [[Simple polygon|simple]] [[isotoxal]] star ☆ without self-intersecting edges. It is often found as a facet inside Islamic [[Girih tiles]], of which there are five different rudimentary types.{{Cite journal |last=Sarhangi |first=Reza |url=https://link.springer.com/content/pdf/10.1007/s00004-012-0117-5.pdf?pdf=button |title=Interlocking Star Polygons in Persian Architecture: The Special Case of the Decagram in Mosaic Designs |journal=Nexus Network Journal |volume=14 |issue=2 |year=2012 |page=350 |doi=10.1007/s00004-012-0117-5 |s2cid=124558613}} Generally, [[Star polyhedron|star polytopes]] that are [[Regular polytope|regular]] only exist in [[dimension]]s

2

⩽

n

5

, and can be constructed using five [[Stellation#Miller's rules|Miller rules]] for stellating polyhedra or higher-dimensional [[polytope]]s.{{Cite book |last1=Coxeter |first1=H. S. M. |last2=du Val |first2=P. |last3=Flather |first3=H. T. |last4=Petrie |first4=J.F. |author1-link =H. S. M. Coxeter |author2-link=Patrick du Val |display-authors=2 |url=https://link.springer.com/book/10.1007/978-1-4613-8216-4 |title=The Fifty-Nine Icosahedra |publisher=[[Springer-Verlag]] |edition=1 |location=New York |year=1982 |pages=7, 8 |doi=10.1007/978-1-4613-8216-4 |isbn=978-0-387-90770-3 |oclc=8667571 |s2cid=118322641 }} [140] => [141] => ==== Graphs theory, and planar geometry ==== [142] => [143] => In [[graph theory]], all [[Graph theory|graphs]] with four or fewer vertices are [[Planar graph|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.{{Cite journal |last=Burnstein |first=Michael|title=Kuratowski-Pontrjagin theorem on planar graphs |journal=[[Journal of Combinatorial Theory]] | series=Series B |volume=24 |issue=2 |year=1978 |pages=228–232 |doi= 10.1016/0095-8956(78)90024-2 |doi-access=free }} A similar graph is the [[Petersen graph]], which is [[strongly connected]] and also [[Planar graph|nonplanar]]. It is most easily described as graph of a pentagram ''embedded'' inside a pentagon, with a total of 5 [[Crossing number (graph theory)|crossings]], a [[Girth (graph theory)|girth]] of 5, and a [[Thue number]] of 5.{{Cite book|first1=D. A. |last1=Holton |first2=J. |last2=Sheehan |title-link= The Petersen Graph |title=The Petersen Graph|publisher=[[Cambridge University Press]]|year=1993|pages=9.2, 9.5 and 9.9|isbn=0-521-43594-3}}{{Cite journal |author1-link= Noga Alon |last1= Alon |first1= Noga|last2= Grytczuk |first2= Jaroslaw |last3= Hałuszczak |first3= Mariusz |last4= Riordan |first4= Oliver |title= Nonrepetitive colorings of graphs |journal= Random Structures & Algorithms |volume= 2 |issue= 3–4 | year= 2002 |page= 337 |doi= 10.1002/rsa.10057 |mr= 1945373 |s2cid= 5724512 |url= http://www.math.tau.ac.il/~nogaa/PDFS/aghr2.pdf |quote=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. }} The Petersen graph, which is also a [[distance-regular graph]], is one of only 5 known [[Connectivity (graph theory)|connected]] [[vertex-transitive]] graphs with no [[Hamiltonian cycle]]s.Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."] {{webarchive|url=https://web.archive.org/web/20080720005020/http://www.cs.uwa.edu.au/~gordon/remote/foster/ |date=2008-07-20 }} The [[automorphism group]] of the Petersen graph is the [[symmetric group]]

\mathrm{S}_{5}

of [[group order|order]] [[120 (number)|120]] = 5!. [144] => [145] => The [[Hadwiger–Nelson problem|chromatic number]] of the [[Plane (geometry)|plane]] is at least five, depending on the choice of [[Axiom of choice|set-theoretical axioms]]: the minimum number of [[Graph coloring|colors]] required to color the plane such that no pair of points at a distance of 1 has the same color.{{Cite journal |last=de Grey |first=Aubrey D.N.J. |author-link=Aubrey de Grey |title=The Chromatic Number of the Plane is At Least 5 |journal=[[Geombinatorics]] |volume=28 |pages=5–18 |year=2018 |arxiv=1804.02385 |mr=3820926 |s2cid=119273214 }}{{Cite journal |last1=Exoo |first1=Geoffrey |last2=Ismailescu |first2=Dan |title=The Chromatic Number of the Plane is At Least 5: A New Proof |url=https://link.springer.com/article/10.1007/s00454-019-00058-1 |journal=[[Discrete & Computational Geometry]] |volume=64 |pages=216–226 |publisher=[[Springer Science+Business Media|Springer]] |location=New York, NY |year=2020 |doi=10.1007/s00454-019-00058-1 |arxiv=1805.00157 |mr=4110534 |s2cid=119266055 |zbl=1445.05040 }} 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 spindle]]s 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 (mathematical logic)|structure]]. [146] => [147] => The plane also contains a total of five [[Bravais lattice]]s, or arrays of [[Point (geometry)|points]] defined by discrete [[Translation (geometry)|translation]] operations: [[Hexagonal lattice|hexagonal]], [[Oblique lattice|oblique]], [[Rectangular lattice|rectangular]], [[Rectangular lattice#Bravais lattices|centered rectangular]], and [[Square lattice|square]] lattices. [[Euclidean tilings by convex regular polygons|Uniform tilings]] of the plane, furthermore, are generated from combinations of only five regular polygons: the [[triangle]], [[square]], [[hexagon]], [[octagon]], and the [[dodecagon]].{{Cite journal |first1=Branko |last1=Grünbaum |author-link=Branko Grünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=[[Mathematics Magazine]] |volume=50 |issue=5 |publisher=Taylor & Francis, Ltd.|pages=227–236 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 }} The plane can also be tiled [[Pentagonal tiling#Monohedral convex pentagonal tilings|monohedrally]] with convex [[pentagonal tiling|pentagons]] in fifteen different ways, three of which have [[Laves tiling]]s as special cases.{{Cite book |last1=Grünbaum |first1=Branko |author-link=Branko Grünbaum |last2=Shephard |first2=Geoffrey C. |title=Tilings and Patterns |location=New York |publisher=W. H. Freeman and Company |year=1987 |isbn=978-0-7167-1193-3 |chapter=Tilings by polygons |mr=0857454 |url-access=registration |url=https://archive.org/details/isbn_0716711931 }} Section 9.3: "Other Monohedral tilings by convex polygons". [148] => [149] => ==== Polyhedra ==== [150] => [151] => [[File:De divina proportione - Illustration 13, crop.jpg|150px|right|thumb|Illustration by [[Leonardo da Vinci]] of a [[regular dodecahedron]], from [[Luca Pacioli]]'s ''[[Divina proportione]]'']] [152] => [153] => There are five [[Platonic solid]]s in [[three-dimensional space]] that are [[Regular polyhedron|regular]]: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 61 The [[Regular dodecahedron|dodecahedron]] in particular contains [[pentagonal]] faces, while the [[Regular icosahedron|icosahedron]], its [[Dual polytope|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 polyhedron|semi-regular]], which are called the [[Archimedean solid]]s. There are also five: [154] => [155] => {{Bullet list [156] => |[[Uniform polyhedron compound|Regular polyhedron compounds]]: the [[compound of five tetrahedra]], compound of ten tetrahedra, compound of five cubes, compound of five octahedra, and [[stella octangula]].{{cite journal |first=John |last=Skilling |title=Uniform Compounds of Uniform Polyhedra |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=79 |pages=447–457 |year=1976 |issue=3 |doi=10.1017/S0305004100052440 |bibcode=1976MPCPS..79..447S |mr=0397554|s2cid=123279687 }} [[Icosahedral symmetry]]

\mathrm I_{h}

is [[isomorphic]] to the [[alternating group]] on five letters

\mathrm A_{5}

of order [[120 (number)|120]], realized by actions on these uniform polyhedron compounds (aside from the regular compound of two tetrahedra). All fifteen [[mirror plane]]s of

\mathrm I_{h}

pass through the edges of a regular [[Spherical polyhedron|spherical]] [[compound of five octahedra]], whose sets of three orthogonal [[great circle]]s use five colors.{{efn|1=[[File:Spherical disdyakis triacontahedron as compound of five octahedra.png|155px]]}}{{Cite conference |editor-last=Sarhangi |editor-first=Reza |author-last=Hart |author-first=George W. |author-link= George W. Hart |title=Icosahedral Constructions |url=http://t.archive.bridgesmathart.org/1998/bridges1998-195.pdf |book-title=Bridges: Mathematical Connections in Art, Music, and Science |publisher=[[The Bridges Organization]] |location=Winfield, Kansas |year=1998 |page=196 |isbn=978-0-9665201-0-1 |oclc=59580549 |s2cid=202679388 }}{{Cite web |last=Hart |first=George W. |author-link= George W. Hart |title=Symmetry Planes |website=Virtual Polyhedra (The Encyclopedia of Polyhedra) |url=https://www.georgehart.com/virtual-polyhedra/symmetry_planes.html |access-date=2023-09-27 }} [157] => :"They can be colored as five sets of three mutually orthogonal planes" where the "fifteen planes divide the sphere into 120 [[Schwarz triangle|Möbius triangle]]s." [158] => |[[Honeycomb (geometry)|Space-filling]] [[convex polyhedron|convex polyhedra]] with regular faces: the triangular prism, [[hexagonal prism]], cube, truncated octahedron, and [[gyrobifastigium]].{{Cite book |title=The Six-Cornered Snowflake |first=Johannes |last=Kepler |author-link=Johannes Kepler |publisher=Paul Dry Books |year=2010 |isbn=978-1-58988-285-0 |at=Footnote 18, [https://books.google.com/books?id=yE8yTUFWLXgC&pg=PA146 p. 146] }} 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 solid]]s, respectively, that can tessellate space with their own copies. [159] => |[[Isohedral figure#Related terms|Cell-transitive]] [[Parallelohedron|parallelohedra]]: any [[parallelepiped]], as well as the [[rhombic dodecahedron]], the [[elongated dodecahedron]], the hexagonal prism and the truncated octahedron.{{cite book|last=Alexandrov|first=A. D.|author-link=Aleksandr Danilovich Aleksandrov|contribution=8.1 Parallelohedra|pages=349–359|publisher=Springer|title=Convex Polyhedra|title-link=Convex Polyhedra (book)|year=2005}} The cube is a special case of a parallelepiped, and the rhombic dodecahedron (with five [[stellation]]s per [[Stellation#Miller's rules|Miller's rules]]) is the only [[Catalan solid]] to tessellate space on its own.{{Cite web |url=https://www.software3d.com/Enumerate.php |last=Webb |first=Robert |title=Enumeration of Stellations |website=www.software3d.com |access-date=2023-01-12 |archive-url=https://archive.today/20221126015207/https://www.software3d.com/Enumerate.php |archive-date=2022-11-26 }} [160] => |[[List of regular polytopes and compounds#Abstract polytopes|Regular abstract polyhedra]], which include the [[excavated dodecahedron]] and the [[dodecadodecahedron]].{{cite journal |last=Wills |first=J. M. |title=The combinatorially regular polyhedra of index 2 |journal=Aequationes Mathematicae |volume=34 |year=1987 |issue=2–3 |pages=206–220 |doi=10.1007/BF01830672 |s2cid=121281276 }} They have combinatorial symmetries transitive on [[List of regular polytopes and compounds#Abstract polytopes|flags]] of their elements, with [[Topology|topologies]] equivalent to that of [[toroid]]s and the ability to tile the [[hyperbolic plane]]. [161] => }} [162] => [163] => Moreover, the fifth [[pentagonal pyramidal number]]

75 = 15 \times 5

represents the total number of indexed ''[[Uniform polyhedron compound|uniform compound polyhedra]]'', which includes seven families of [[Prism (geometry)|prisms]] and [[antiprism]]s. Seventy-five is also the number of non-[[Prism (geometry)|prismatic]] [[uniform polyhedra]], which includes Platonic solids, Archimedean solids, and [[Star polyhedron|star polyhedra]]; there are also precisely ''five'' [[uniform prism]]s and [[antiprism]]s that contain pentagons or pentagrams as faces — the [[pentagonal prism]] and [[Pentagonal antiprism|antiprism]], and the [[pentagrammic prism]], [[Pentagrammic antiprism|antiprism]], and [[Pentagrammic crossed antiprism|crossed-antirprism]].{{Cite journal |last=Har'El |first=Zvi |url=http://harel.org.il/zvi/docs/uniform.pdf |title=Uniform Solution for Uniform Polyhedra |journal=[[Geometriae Dedicata]] |volume=47 |pages=57–110 |publisher=[[Springer Publishing]] |location=Netherlands |year=1993 |doi=10.1007/BF01263494 |mr=1230107 |zbl=0784.51020 |s2cid=120995279 }} [164] => :"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 triangle]]s."
Appendix II: Uniform Polyhedra. In all, there are twenty-five [[Uniform polyhedron|uniform polyhedra]] that generate four-dimensional [[Uniform polychoron|uniform polychora]], they are the five Platonic solids, fifteen Archimedean solids counting two [[Chirality (mathematics)|enantiomorphic]] forms, and five associated prisms: the [[Triangular prism|triangular]], [[Pentagonal prism|pentagonal]], [[Hexagonal prism|hexagonal]], [[Octagonal prism|octagonal]], and [[Decagonal prism|decagonal]] prisms. [165] => [166] => ==== Fourth dimension ==== [167] => [168] => [[File:Schlegel wireframe 5-cell.png|150px|thumb|The four-dimensional [[5-cell]] is the simplest regular [[polychoron]].]] [169] => [170] => The [[5-cell|pentatope]], or 5-cell, is the self-dual fourth-dimensional analogue of the [[tetrahedron]], with [[Coxeter group]] symmetry

\mathrm{A}_{4}

of order [[120 (number)|120]] = 5[[Factorial|!]] and

\mathrm{S}_{5}

[[Mathematical structure|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 [[Uniform 4-polytope|regular 4-polytopes]], made of thirty-one [[Simplex#Elements|elements]]: five [[Vertex (geometry)|vertices]], ten [[Edge (geometry)|edges]], ten [[Face (geometry)|faces]], five [[Regular tetrahedron|tetrahedral cells]] and one [[Face (geometry)#Facet or (n − 1)-face|4-face]].{{Cite book |author=H. S. M. Coxeter |author-link=Harold Scott MacDonald Coxeter |title=[[Regular Polytopes (book)|Regular Polytopes]] |publisher=[[Dover Publications, Inc.]] |edition=3rd |year=1973 |location=New York |pages=1–368 |isbn=978-0-486-61480-9 }}{{rp|p.120}} [171] => [172] => {{Bullet list [173] => |A [[120-cell|regular 120-cell]], the dual ''polychoron'' to the regular [[600-cell]], can fit one hundred and twenty 5-cells. Also, five [[24-cell]]s 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 polytope|uniform]] nonconvex [[duoantiprism|''duoantiprismatic'']] solution in the fourth dimension, constructed from the [[polytope]] [[cartesian product]]

\{5\}\otimes\{5/3\}

and made of fifty [[tetrahedra]], ten [[pentagrammic crossed antiprism]]s, ten [[pentagonal antiprism]]s, and fifty vertices.{{rp|p.124}} [174] => [175] => |The [[grand antiprism]], which is the only known [[Wythoff construction|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.{{Cite book |author1=John Horton Conway |author1-link= John Horton Conway |author2=Heidi Burgiel |author3=Chaim Goodman-Strauss |url=https://www.routledge.com/The-Symmetries-of-Things/Conway-Burgiel-Goodman-Strauss/p/book/9781568812205 |title=The Symmetries of Things |publisher=[[A K Peters]]/CRC Press |year=2008 |isbn=978-1-56881-220-5 }} Chapter 26: "The Grand Antiprism" [176] => |The [[Abstract polytope|abstract]] four-dimensional [[57-cell]] is made of fifty-seven [[Hemi-icosahedron|hemi-icosahedral]] cells, in-which five surround each edge.{{Cite journal |last = Coxeter |first = H. S. M. |author-link = Harold Scott MacDonald Coxeter |doi = 10.1007/BF00149428 |issue = 1 |journal = [[Geometriae Dedicata]] |mr = 679218 |pages = 87–99 |title = Ten toroids and fifty-seven hemidodecahedra |volume = 13 |year = 1982 |s2cid = 120672023 }}. The [[11-cell]], another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven [[Hemi-dodecahedron|hemi-dodecahedral cells]] each with fifteen edges.{{Cite journal |last=Coxeter |first=H. S. M |author-link = Harold Scott MacDonald Coxeter |url=https://www.sciencedirect.com/science/article/abs/pii/S0304020808728147 |title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra |journal=Annals of Discrete Mathematics |series=North-Holland Mathematics Studies |issue=20 |year=1984 |volume=87 |pages=103–114 |doi=10.1016/S0304-0208(08)72814-7 |isbn=978-0-444-86571-7 }} The [[n-skeleton|skeleton]] of the hemi-dodecahedron is the [[Petersen graph]]. [177] => }} [178] => [179] => Overall, the fourth dimension contains five fundamental [[Uniform 4-polytope#Convex uniform 4-polytopes|Weyl groups]] that form a finite number of [[Uniform 4-polytope#Enumeration|uniform polychora]] based on only twenty-five uniform polyhedra: [[A4 polytope|

\mathrm A_{4}

]], [[B4 polytope|

\mathrm B_{4}

]], [[D4 polytope|

\mathrm D_{4}

]], [[F4 polytope|

\mathrm F_{4}

]], and [[H4 polytope|

\mathrm H_{4}

]], accompanied by a fifth or sixth general group of unique [[Uniform 4-polytope#Prismatic uniform 4-polytopes|4-prisms]] of Platonic and Archimedean solids. There are also a total of five [[Coxeter group]]s that generate non-prismatic [[Uniform 5-polytope#Regular and uniform honeycombs|Euclidean honeycombs]] in 4-space, alongside five [[Uniform 5-polytope#Regular and uniform hyperbolic honeycombs|compact hyperbolic Coxeter groups]] that generate five regular [[Uniform 5-polytope#Compact regular tessellations of hyperbolic 4-space|compact hyperbolic honeycombs]] with finite [[Facet (geometry)|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 [[Coxeter-Dynkin diagram#Ranks 4–5|rank 5]], with [[Coxeter–Dynkin diagram#Paracompact (Koszul simplex groups)|paracompact hyperbolic solutions]] existing through rank 10. Likewise, analogues of four-dimensional

\mathrm{H}_{4}

[[H4 polytope|hexadecachoric]] or

\mathrm{F}_{4}

[[List of F4 polytopes|icositetrachoric]] symmetry do not exist in dimensions

n

⩾

5

; however, there are [[Uniform 5-polytope#Uniform prismatic forms|prismatic groups]] in the fifth dimension which contains [[Prism (geometry)|prisms]] of regular and uniform [[Uniform 4-polytope|4-polytopes]] that have

\mathrm{H}_{4}

and

\mathrm{F}_{4}

symmetry. There are also five regular [[List of regular polytopes and compounds#Regular projective 4-polytopes|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.{{Cite book |last1= McMullen |first1= Peter |author1-link= Peter McMullen |last2= Schulte |first2= Egon |author2-link= Egon Schulte |url = https://archive.org/details/abstractregularp0000mcmu |url-access= registration |title= Abstract Regular Polytopes |publisher= Cambridge University Press |location= Cambridge |series= Encyclopedia of Mathematics and its Applications |volume= 92 |year= 2002 |pages=162–164 |doi = 10.1017/CBO9780511546686 |isbn= 0-521-81496-0 |mr= 1965665 |s2cid=115688843 }} Only two regular projective polytopes exist in each higher dimensional space. [180] => [181] => [[File:Fundamental domain of Bring curve.svg|160px|thumb|The [[fundamental polygon]] for [[Bring's curve]] is a regular [[Hyperbolic geometry|hyperbolic]] twenty-sided [[icosagon]].]] [182] => [183] => In particular, [[Bring's curve|Bring's surface]] is the curve in the [[Projective space|projective plane]]

\mathbb{P}^4

that is represented by the [[homogeneous equation]]s:{{Cite journal |last1=Edge |first1=William L. |author-link=William Edge (mathematician)|title=Bring's curve |url=https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-18.3.539 |journal=[[Journal of the London Mathematical Society]] |publisher=[[London Mathematical Society]] |location=London |year=1978 |volume=18 |issue=3 |pages=539–545 |doi=10.1112/jlms/s2-18.3.539 |issn=0024-6107 |mr=518240 |s2cid=120740706 |zbl=0397.51013 }} [184] => [185] => :

v+w+x+y+z=v^2+w^2+x^2+y^2+z^2=v^3+w^3+x^3+y^3+z^3=0.

[186] => [187] => It holds the largest possible [[automorphism group]] of a [[Genus (mathematics)|genus]] four [[complex curve]], with group structure

\mathrm S_{5}

. This is the [[Riemann surface]] associated with the [[small stellated dodecahedron]], whose [[fundamental polygon]] is a regular hyperbolic [[icosagon]], with an area of

12\pi

(by the [[Gauss-Bonnet theorem]]). Including reflections, its full group of symmetries is

\mathrm S_{5} \times \mathbb{Z}_{2}

, of order [[240 (number)|240]]; which is also the number of {{math|(2,4,5)}} hyperbolic triangles that tessellate its fundamental polygon. [[Bring radical|Bring quintic]]

x^5+ax+b = 0

holds roots

x_{i}

that satisfy Bring's curve. [188] => [189] => ==== Fifth dimension ==== [190] => [191] => The [[5-simplex]] or ''hexateron'' is the [[Five-dimensional space|five-dimensional]] analogue of the 5-cell, or 4-simplex. It has Coxeter group

\mathrm{A}_{5}

as its symmetry group, of order 720 = 6[[Factorial|!]], whose group structure is represented by the symmetric group

\mathrm{S}_{6}

, the only finite symmetric group which has an [[outer automorphism]]. The [[5-cube]], made of ten [[tesseract]]s and the 5-cell as its vertex figure, is also regular and one of thirty-one [[uniform 5-polytope]]s under the Coxeter [[Uniform 5-polytope#The B5 family|

\mathrm B_{5}

hypercubic]] group. The [[demipenteract]], with one hundred and twenty [[Cell (geometry)|cells]], is the only fifth-dimensional [[Semiregular polytope|semi-regular polytope]], and has the [[rectified 5-cell]] as its vertex figure, which is one of only three semi-regular 4-polytopes alongside the [[rectified 600-cell]] and the [[snub 24-cell]]. In the fifth dimension, there are five regular paracompact honeycombs, all with [[Infinity|infinite]] [[Facet (geometry)|facets]] and [[vertex figure]]s; no other regular paracompact honeycombs exist in higher dimensions.{{cite web|author=H.S.M. Coxeter|url=https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.361.251|title=Regular Honeycombs in Hyperbolic Space |year=1956|page=168|citeseerx=10.1.1.361.251 }} There are also exclusively twelve [[Complex polytope#Regular complex 5-apeirotopes and higher|complex aperiotopes]] in [[Complex coordinate space|

\mathbb{C}^n

complex spaces]] of dimensions

n

⩾

5

; alongside [[Complex polytope#Enumeration of regular complex 5-polytopes|complex polytopes]] in

\mathbb{C}^5

and higher under [[simplex]], [[hypercube|hypercubic]] and [[orthoplex]] groups (with [[Complex polytope#van Oss polygon|van Oss polytopes]]).{{Cite book |author=H. S. M. Coxeter |author-link=H. S. M. Coxeter |title=Regular Complex Polytopes |publisher=Cambridge University Press |edition=2nd |year=1991 |pages=144–146 |doi=10.2307/3617711 |jstor=3617711 |isbn=978-0-521-39490-1 |s2cid=116900933 |zbl=0732.51002 }} [192] => [193] => A [[Veronese surface]] in the [[Projective space|projective plane]]

\mathbb{P}^5

generalizes a [[Pairing|linear]] condition

\nu:\mathbb{P}^2\to \mathbb{P}^5

for a point to be contained inside a ''[[conic]]'', which [[Five points determine a conic|requires five points in the same way]] that two points are needed to determine a [[Line (geometry)|line]].{{Cite journal |first = A. C. |last = Dixon |author-link = Alfred Cardew Dixon |title = The Conic through Five Given Points |journal = The Mathematical Gazette |volume = 4 |number = 70 |date=March 1908 |pages = 228–230 |publisher = The Mathematical Association |jstor = 3605147 |doi = 10.2307/3605147 |s2cid = 125356690 |url = https://zenodo.org/record/2014634 }} [194] => [195] => ==== Finite simple groups ==== [196] => [197] => There are five complex [[exceptional Lie algebra]]s: [[G2 (mathematics)|

\mathfrak{g}_2

]], [[F4 (mathematics)|

\mathfrak{f}_4

]], [[E6 (mathematics)|

\mathfrak{e}_6

]], [[E7 (mathematics)|

\mathfrak{e}_7

]], and [[E8 (mathematics)|

\mathfrak{e}_8

]]. The [[Faithful representation|smallest]] of these,

\mathfrak{g}_2

of [[Real coordinate space|real]] dimension 28, can be represented in five-dimensional complex space and [[Projective geometry|projected]] as a [[Ball (geometry)|ball]] rolling on top of another ball, whose [[motion]] is described in two-dimensional space.{{Cite journal|title = G₂ and the rolling ball |last1=Baez |first1=John C. |author1-link=John C. Baez|last2=Huerta |first2=John |journal=Trans. Amer. Math. Soc. |volume=366 |issue=10 |year=2014 |pages=5257–5293 |doi=10.1090/s0002-9947-2014-05977-1 |mr=3240924 |s2cid=50818244 |doi-access=free }}

\mathfrak{e}_8

is the largest, and holds the other four Lie algebras as [[subgroup]]s, with a representation over

\mathbb {R}

in dimension 496. It contains an associated [[E8 lattice|lattice]] that is constructed with one hundred and twenty quaternionic [[Icosian|unit icosians]] that make up the vertices of the [[600-cell]], whose Euclidean [[Quaternion#Conjugation, the norm, and reciprocal|norms]] define a quadratic form on a lattice structure [[Isomorphism|isomorphic]] to the optimal configuration of spheres in eight dimensions.{{Cite journal |last=Baez |first=John C. |author-link=John C. Baez |title=From the Icosahedron to E₈ |journal=London Math. Soc. Newsletter |volume=476 |pages=18–23 |year=2018 |arxiv=1712.06436 |mr=3792329 |s2cid=119151549 |zbl=1476.51020 }} This [[Sphere packing problem|sphere packing]]

\mathrm {E}_{8}

lattice structure in [[Uniform 8-polytope|8-space]] is held by the vertex arrangement of the [[5 21 honeycomb|'''5₂₁''' honeycomb]], one of five Euclidean honeycombs that admit [[Thorold Gosset|Gosset's]] original definition of a [[Semiregular polytope|semi-regular honeycomb]], which includes the three-dimensional [[Tetrahedral-octahedral honeycomb|alternated cubic honeycomb]].{{Cite journal |author=H. S. M. Coxeter |author-link=H. S. M. Coxeter |title=Seven Cubes and Ten 24-Cells |url=https://link.springer.com/content/pdf/10.1007/PL00009338.pdf |journal=[[Discrete & Computational Geometry]] |volume=19 |year=1998 |issue=2 |pages=156–157 |doi=10.1007/PL00009338 |doi-access=free |zbl=0898.52004 |s2cid=206861928 }}{{Cite journal |author=Thorold Gosset |author-link=Thorold Gosset |title=On the regular and semi-regular figures in space of n dimensions |url=https://www.maths.ed.ac.uk/~v1ranick/papers/gosset.pdf |journal=Messenger of Mathematics |volume=29 |year=1900 |pages=43–48 |jfm=30.0494.02 }} The smallest simple isomorphism found inside finite simple [[Lie group]]s is

\mathrm {A_{5}} \cong A_{1}(4) \cong A_{1}(5)

,{{Cite book |last1=Conway |first1=J. H. |author1-link=John Horton Conway |last2=Curtis |first2=R. T. |last3=Norton |first3=S. P. |author3-link=Simon P. Norton |last4=Parker |first4=R. A. |author4-link=Richard A. Parker |last5=Wilson |first5=R. A. |author5-link=Robert Arnott Wilson |title=[[ATLAS of Finite Groups|ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups]] |publisher=[[Clarendon Press]] |page=xv |year=1985 |location=Oxford |isbn=978-0-19-853199-9 |oclc=12106933 |mr=827219 |s2cid=117473588 |zbl=0568.20001 }} where here

\mathrm {A_{n}}

represents [[alternating group]]s and

A_{n}(q)

[[Group of Lie type#Chevalley groups|classical Chevalley groups]]. In particular, the smallest non-solvable group is the alternating group on five letters, which is also the smallest [[Simple group|simple]] non-[[Abelian group|abelian]] group. [198] => [199] => The five [[Mathieu groups]] constitute the [[Sporadic group#First generation (5 groups): the Mathieu groups|first generation]] in the [[Sporadic groups#Happy Family|happy family]] of [[sporadic groups]]. These are also the first five sporadic groups [[Classification of finite simple groups#Timeline of the proof|to have been described]], defined as

\mathrm{M}_{n}

[[Mathieu groups#Multiply transitive groups|multiply transitive]] permutation groups on

n

[[Group object|objects]], with

n

[[Element (mathematics)|∈]] {11, 12, 22, 23, 24}.{{Cite book |author=Robert L. Griess, Jr. |author-link=Robert Griess |url=https://link.springer.com/book/10.1007/978-3-662-03516-0 |title=Twelve Sporadic Groups |series=Springer Monographs in Mathematics |publisher=Springer-Verlag |location=Berlin |year=1998 |pages=1−169 |isbn=978-3-540-62778-4 |doi=10.1007/978-3-662-03516-0 |mr=1707296 |s2cid=116914446 |zbl=0908.20007 }}{{rp|p.54}} In particular,

\mathrm{M}_{11}

, the smallest of all sporadic groups, has a [[rank 3 action]] on fifty-five points from an [[Induced representation|induced action]] on [[unordered pair]]s, as well as two [[five-dimensional space|five-dimensional]] [[Faithful representation|faithful complex irreducible representations]] over the [[Field (mathematics)|field]] with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with

n

elements.{{Cite journal |last=Jansen |first=Christoph |date=2005 |title=The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups |journal=[[LMS Journal of Computation and Mathematics]] |volume=8 |pages=123–124 |publisher=[[London Mathematical Society]] |doi=10.1112/S1461157000000930 |doi-access=free |mr=2153793 |s2cid=121362819 |zbl=1089.20006 }} Of precisely five different [[conjugacy class]]es of [[maximal subgroup]]s of

\mathrm{M}_{11}

, one is the [[Almost simple group|almost simple]] symmetric group [[Symmetric group#Low degree groups|

\mathrm{S}_5

]] (of order 5[[Factorial|!]]), and another is

\mathrm{M}_{10}

, also almost simple, that functions as a [[point stabilizer]] which contains five as its largest [[prime factor]] in its [[group order]]: {{math|1=2⁴·3²·5 = [[2]]·[[3]]·[[4]]·5·[[6]] = [[8]]·[[9]]·[[10]] = 720}}. On the other hand, whereas

\mathrm{M}_{11}

is sharply 4-transitive,

\mathrm{M}_{12}

is [[Mathieu groups#Multiply transitive groups|sharply 5-transitive]] and

\mathrm{M}_{24}

is 5-transitive, and as such they are the only two 5-transitive groups that are not [[symmetric group]]s or [[alternating group]]s.{{Cite book |last=Cameron |first=Peter J. |title=Projective and Polar Spaces |chapter=Chapter 9: The geometry of the Mathieu groups |chapter-url=https://webspace.maths.qmul.ac.uk/p.j.cameron/pps/pps9.pdf |publisher=University of London, Queen Mary and Westfield College |year=1992 |page=139|isbn=978-0-902-48012-4 |s2cid=115302359 }}

\mathrm{M}_{22}

has the first five prime numbers as its distinct prime factors in its order of {{math|1=2⁷·3²·5·[[7]]·[[11 (number)|11]]}}, and is the smallest of five sporadic groups with five distinct prime factors in their order.{{rp|p.17}} All Mathieu groups are subgroups of

\mathrm{M}_{24}

, which under the [[Witt design]]

\mathrm{W}_{24}

of [[Steiner system#The Steiner system S(5, 8, 24)|Steiner system]]

\operatorname{S(5, 8, 24)}

emerges a construction of the [[Binary Golay code|extended binary Golay code]]

\mathrm{B}_{24}

that has

\mathrm{M}_{24}

as its [[automorphism group]].{{rp|pp.39,47,55}}

\mathrm{W}_{24}

generates ''octads'' from [[Code word (communication)|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.{{rp|p.38}} 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 primarily constructed using the [[Weyl character formula#Statement of Weyl character formula|Weyl vector]]

(0,1,2,3, \dots ,24; 70)

that admits the only non-unitary solution to the ''[[cannonball problem]]'', where the sum of the [[Square number|squares]] of the first twenty-four integers is equivalent to the square of another integer, the fifth [[pentatope number]] (70). The [[subquotient]]s of the automorphism of the Leech lattice, [[Conway group]]

\mathrm{Co}_{0}

, is in turn the subject of the [[Sporadic group#Second generation (7 groups): the Leech lattice|second generation]] of seven sporadic groups.{{rp|pp.99,125}} [200] => [201] => There are five ''non-supersingular'' prime numbers — [[37 (number)|37]], [[43 (number)|43]], [[53 (number)|53]], [[61 (number)|61]], and [[67 (number)|67]] — less than [[71 (number)|71]], which is the largest of fifteen [[Supersingular prime (moonshine theory)|supersingular primes]] that divide the [[Order (group theory)|order]] of the ''[[Monster group|friendly giant]]'', itself the largest sporadic group.{{Cite journal |author=Luis J. Boya |title=Introduction to Sporadic Groups |journal=Symmetry, Integrability and Geometry: Methods and Applications |page=13 |date=2011-01-16 |volume=7 |doi=10.3842/SIGMA.2011.009 |arxiv=1101.3055 |bibcode=2011SIGMA...7..009B |s2cid=16584404 }} In particular, a [[Centralizer and normalizer|centralizer]] of an element of order 5 inside this group arises from the product between [[Harada–Norton group|Harada–Norton]] sporadic group

\mathrm{HN}

and a group of order 5.{{Cite journal |last1=Lux |first1=Klaus |last2=Noeske |first2=Felix |last3=Ryba |first3=Alexander J. E. |title=The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2 |journal=[[Journal of Algebra]] |publisher=[[Elsevier]] |volume=319 |issue=1 |year=2008 |location=Amsterdam |pages=320–335 |doi=10.1016/j.jalgebra.2007.03.046 |doi-access=free |mr=2378074 |s2cid=120706746 |zbl=1135.20007 }}{{Cite journal |last=Wilson |first=Robert A. |author-link=Robert Arnott Wilson |title=The odd local subgroups of the Monster |journal=Journal of Australian Mathematical Society (Series A) |publisher=[[Cambridge University Press]] |volume=44 |issue=1 |pages=12–13 |year=2009 |location=Cambridge |doi=10.1017/S1446788700031323 |doi-access=free |mr=914399 |s2cid=123184319 |zbl=0636.20014 }} On its own,

\mathrm{HN}

can be represented using [[Generator (mathematics)|standard generators]]

(a,b,ab)

that further dictate a condition where

o([a, b]) = 5

.{{Cite book |last=Wilson |first=R.A |author-link=Robert Arnott Wilson |title=The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249) |chapter=Chapter: An Atlas of Sporadic Group Representations |chapter-url=https://webspace.maths.qmul.ac.uk/r.a.wilson/pubs_files/ASGRweb.pdf|publisher=Cambridge University Press |location=Cambridge |year=1998 |page=267 |doi=10.1017/CBO9780511565830.024 |isbn=978-0-511-56583-0 |oclc=726827806 |zbl=0914.20016 |s2cid=59394831 }}{{Cite journal |last1=Nickerson |first1=S.J. |last2=Wilson |first2=R.A. |author2-link=Robert Anton Wilson |title=Semi-Presentations for the Sporadic Simple Groups |url=https://www.tandfonline.com/doi/abs/10.1080/10586458.2005.10128927 |journal=Experimental Mathematics |volume=14 |issue=3 |page=367 |publisher=[[Taylor & Francis]] |year=2011 |location=Oxfordshire |doi=10.1080/10586458.2005.10128927 |mr=2172713 |zbl=1087.20025 |s2cid=13100616 }} This condition is also held by other generators that belong to the [[Tits group]]

\mathrm{T}

,{{Cite web |last1=Wilson |first1=R.A. |author1-link=Robert Arnott Wilson |last2=Parker |first2=R.A. |author2-link=Richard A. Parker |last3=Nickerson |first3=S.J. |last4=Bray |first4=J.N. |title=Exceptional group ²F₄(2)', Tits group T |url=https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TF42/ |website=ATLAS of Finite Group Representations |year=1999 }} 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,

\mathrm{HN}

holds a 133-dimensional representation where 5 acts on a [[Commutative property|commutative]] yet non-[[Associative property|associative]] product as a 5-[[Modular arithmetic|modular]] analogue of the [[Griess algebra]]

V_{2}

^{{{large|♮}}},{{Cite journal |last1=Ryba |first1=A. J. E. |title=A natural invariant algebra for the Harada-Norton group |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |publisher=[[Cambridge University Press]] |volume=119 |issue=4 |location=Cambridge |year=1996 |pages=597–614 |doi=10.1017/S0305004100074454 |bibcode=1996MPCPS.119..597R |mr=1362942 |s2cid=119931824 |zbl=0851.20034 }} which holds the friendly giant as its [[automorphism group]]. [202] => [203] => === Euler's identity === [204] => [205] => [[Euler's identity]],

e^{i\pi}

1

0

, contains five essential [[number]]s used widely in mathematics: [[Pi (mathematical constant)|Archimedes' constant]]

\pi

, [[e (mathematical constant)|Euler's number]]

e

, the [[imaginary number]]

i

, [[Unity (mathematics)|unity]]

1

, and [[zero]]

0

.{{Cite book |first= Robin |last= Wilson |author-link= Robin Wilson (mathematician) |title= Euler's Pioneering Equation: The most beautiful theorem in mathematics |publisher= [[Oxford University Press]] |location=Oxford, UK |year= 2018 |isbn= 978-0-192-51406-6 |oclc=990970269 }}{{Cite book |last=Paulos |first=John Allen |author-link= John Allen Paulos |title=Beyond Numeracy: An Uncommon Dictionary of Mathematics |publisher=[[Penguin Books]] |location=New York, NY |year=1992 |page=117 |isbn=0-14-014574-5 |oclc=26361981 }}{{Cite news |last=Gallagher |first=James |title=Mathematics: Why the brain sees maths as beauty |url=https://www.bbc.co.uk/news/science-environment-26151062 |work=[[BBC News Online]] |publisher=British Broadcasting Corporation (BBC) |date=13 February 2014 |access-date=2023-06-02 }} [206] => [207] => === List of basic calculations === [208] => {| class="wikitable" style="text-align: center; background: white" [209] => |- [210] => ! style="width:105px;"|[[Multiplication]] [211] => !1 [212] => !2 [213] => !3 [214] => !4 [215] => !5 [216] => !6 [217] => !7 [218] => !8 [219] => !9 [220] => !10 [221] => !11 [222] => !12 [223] => !13 [224] => !14 [225] => !15 [226] => !16 [227] => !17 [228] => !18 [229] => !19 [230] => !20 [231] => |- [232] => |'''5 × ''x''''' [233] => |'''5''' [234] => |[[10 (number)|10]] [235] => |[[15 (number)|15]] [236] => |[[20 (number)|20]] [237] => |[[25 (number)|25]] [238] => |[[30 (number)|30]] [239] => |[[35 (number)|35]] [240] => |[[40 (number)|40]] [241] => |[[45 (number)|45]] [242] => |[[50 (number)|50]] [243] => |[[55 (number)|55]] [244] => |[[60 (number)|60]] [245] => |[[65 (number)|65]] [246] => |[[70 (number)|70]] [247] => |[[75 (number)|75]] [248] => |[[80 (number)|80]] [249] => |[[85 (number)|85]] [250] => |[[90 (number)|90]] [251] => |[[95 (number)|95]] [252] => |[[100 (number)|100]] [253] => |} [254] => [255] => {| class="wikitable" style="text-align: center; background: white" [256] => |- [257] => ! style="width:105px;"|[[Division (mathematics)|Division]] [258] => !1 [259] => !2 [260] => !3 [261] => !4 [262] => !5 [263] => !6 [264] => !7 [265] => !8 [266] => !9 [267] => !10 [268] => ! style="width:5px;"| [269] => !11 [270] => !12 [271] => !13 [272] => !14 [273] => !15 [274] => |- [275] => |'''5 ÷ ''x''''' [276] => |'''5''' [277] => |2.5 [278] => |1.{{overline|6}} [279] => |1.25 [280] => |rowspan="2"|[[1]] [281] => |0.8{{overline|3}} [282] => |0.{{overline|714285}} [283] => |0.625 [284] => |0.{{overline|5}} [285] => |0.5 [286] => ! [287] => |0.{{overline|45}} [288] => |0.41{{overline|6}} [289] => |0.{{overline|384615}} [290] => |0.3{{overline|571428}} [291] => |0.{{overline|3}} [292] => |- [293] => |'''''x'' ÷ 5''' [294] => |0.2 [295] => |0.4 [296] => |0.6 [297] => |0.8 [298] => |1.2 [299] => |1.4 [300] => |1.6 [301] => |1.8 [302] => |[[2]] [303] => ! [304] => |2.2 [305] => |2.4 [306] => |2.6 [307] => |2.8 [308] => |[[3]] [309] => |} [310] => [311] => {| class="wikitable" style="text-align: center; background: white" [312] => |- [313] => ! style="width:105px;"|[[Exponentiation]] [314] => !1 [315] => !2 [316] => !3 [317] => !4 [318] => !5 [319] => !6 [320] => !7 [321] => !8 [322] => !9 [323] => !10 [324] => ! style="width:5px;"| [325] => !11 [326] => !12 [327] => !13 [328] => !14 [329] => !15 [330] => |- [331] => |'''5{{sup|''x''}}''' [332] => |'''5''' [333] => |[[25 (number)|25]] [334] => |[[125 (number)|125]] [335] => |[[625 (number)|625]] [336] => |rowspan="2"|3125 [337] => |15625 [338] => |78125 [339] => |390625 [340] => |1953125 [341] => |9765625 [342] => ! [343] => |48828125 [344] => |244140625 [345] => |1220703125 [346] => |6103515625 [347] => |30517578125 [348] => |- [349] => |'''''x''{{sup|5}}''' [350] => |[[1]] [351] => |[[32 (number)|32]] [352] => |[[243 (number)|243]] [353] => |[[1024 (number)|1024]] [354] => |7776 [355] => |16807 [356] => |32768 [357] => |59049 [358] => |100000 [359] => ! [360] => |161051 [361] => |248832 [362] => |371293 [363] => |537824 [364] => |759375 [365] => |} [366] => [367] => ==== In decimal ==== [368] => [369] => All multiples of 5 will end in either 5 or {{num|0}}, and [[Fraction (mathematics)#Vulgar, proper, and improper fractions|vulgar fractions]] with 5 or {{num|2}} in the [[fraction (mathematics)|denominator]] do not yield infinite [[decimal]] expansions because they are prime factors of [[10]], the base. [370] => [371] => In the [[Power (mathematics)|powers]] of 5, every power ends with the number five, and from 5³ onward, if the exponent is [[Parity (mathematics)|odd]], then the hundreds digit is [[1]], and if it is even, the hundreds digit is [[6]]. [372] => [373] => A number

n

raised to the fifth power always ends in the same digit as

n

. [374] => [375] => == Science == [376] => [377] => === Astronomy === [378] => *There are five [[Lagrangian point]]s in a two-body system. [379] => [380] => === Biology === [381] => *There are usually considered to be [[five senses]] (in [[Perception#Types of perception|general terms]]); the five basic [[taste]]s are [[sweetness|sweet]], [[taste#Saltiness|salty]], [[taste#Sourness|sour]], [[taste#Bitterness|bitter]], and [[umami]].{{Cite book|last=Marcus|first=Jacqueline B.|url=https://books.google.com/books?id=p2j3v6ImcX0C&q=five+basic+tastes&pg=PT67|title=Culinary Nutrition: The Science and Practice of Healthy Cooking|date=2013-04-15|publisher=Academic Press|isbn=978-0-12-391883-3|page=55|language=en|quote=There are five basic tastes: sweet, salty, sour, bitter and umami...}} [382] => *Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.{{citation |title=Vertebrates: Structures and Functions |series=Biological Systems in Vertebrates |first=S. M. |last=Kisia |publisher=CRC Press |year=2010 |isbn=978-1-4398-4052-8 |page=106 |url=https://books.google.com/books?id=Hl_JvHqOwoIC&pg=PA106 |quote=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.}} [383] => *Five is the number of appendages on most [[sea star|starfish]], which exhibit [[symmetry (biology)#Pentamerism|pentamerism]].{{Cite book|last1=Cinalli|first1=G.|url=https://books.google.com/books?id=hVhBAAAAQBAJ&q=starfish+five+appendages&pg=PA19|title=Pediatric Hydrocephalus|last2=Maixner|first2=W. J.|last3=Sainte-Rose|first3=C.|date=2012-12-06|publisher=Springer Science & Business Media|isbn=978-88-470-2121-1|page=19|language=en|quote=The five appendages of the starfish are thought to be homologous to five human buds}} [384] => [385] => === Computing === [386] => *5 is the [[ASCII]] code of the [[Enquiry character]], which is abbreviated to ENQ.{{Cite book|last=Pozrikidis|first=Constantine|url=https://books.google.com/books?id=qD3SBQAAQBAJ&q=5+is+the+ASCII+code+of+the+Enquiry+character,&pg=PA209|title=XML in Scientific Computing|date=2012-09-17|publisher=CRC Press|isbn=978-1-4665-1228-3|page=209|language=en|quote=5 5 005 ENQ (enquiry)}} [387] => [388] => == Literature == [389] => [390] => === Poetry === [391] => A [[pentameter]] is verse with five repeating feet per line; the [[iambic pentameter]] was the most prominent form used by [[William Shakespeare]].{{Cite book|last1=Veith (Jr.)|first1=Gene Edward|url=https://books.google.com/books?id=4dY0hHqQ4lgC&q=Pentameter+is+verse+with+five+repeating+feet&pg=PA52|title=Omnibus IV: The Ancient World|last2=Wilson|first2=Douglas|date=2009|publisher=Veritas Press|isbn=978-1-932168-86-0|page=52|language=en|quote=The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter)}} [392] => [393] => == Music == [394] => *Modern musical notation uses a [[staff (music)|musical staff]] made of five horizontal lines.{{Cite web|title=STAVE {{!}} meaning in the Cambridge English Dictionary|url=https://dictionary.cambridge.org/dictionary/english/stave|access-date=2020-08-02|website=dictionary.cambridge.org|language=en|quote=the five lines and four spaces between them on which musical notes are written}} [395] => *A scale with five notes per octave is called a [[pentatonic scale]].{{Cite book|last=Ricker|first=Ramon|url=https://books.google.com/books?id=4pIIfRuiRpcC&q=Pentatonic+scale&pg=PA2|title=Pentatonic Scales for Jazz Improvisation|date=1999-11-27|publisher=Alfred Music|isbn=978-1-4574-9410-9|page=2|language=en|quote=Pentatonic scales, as used in jazz, are five note scales}} [396] => *A [[perfect fifth]] is the most consonant harmony, and is the basis for most western tuning systems.{{Cite book|last=Danneley|first=John Feltham|url=https://books.google.com/books?id=Q9gGAAAAQAAJ&q=perfect+fifth+is+the+most+consonant+harmony,&pg=PP107|title=An Encyclopaedia, Or Dictionary of Music ...: With Upwards of Two Hundred Engraved Examples, the Whole Compiled from the Most Celebrated Foreign and English Authorities, Interspersed with Observations Critical and Explanatory|date=1825|publisher=editor, and pub.|language=en|quote=are the perfect fourth, perfect fifth, and the octave}} [397] => *In [[harmonic]]s, the fifth [[harmonic series (music)|partial]] (or 4th [[overtone]]) of a [[fundamental frequency|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 and minor|major]] [[Triad (music)|triad]] [[chord (music)|chord]] when played in [[just intonation]] (most often the case in [[a cappella]] vocal ensemble singing), will contain such a pure major third. [398] => *Five is the lowest possible number that can be the top number of a [[time signature]] with an asymmetric [[meter (music)|meter]]. [399] => [400] => == Religion and mysticism == [401] => [402] => === Judaism === [403] => *The [[Book of Numbers]] is one of five books in the [[Torah]]; the others being the books of [[Book of Genesis|Genesis]], [[Book of Exodus|Exodus]], [[Book of Leviticus|Leviticus]], and [[Book of Deuteronomy|Deuteronomy]]. [404] => :They are collectively called the Five Books of [[Moses]], the Pentateuch ([[Greek language|Greek]] for "five containers", referring to the scroll cases in which the books were kept), or [[Chumash (Judaism)|Humash]] ({{lang|he|חומש}}, [[Hebrew language|Hebrew]] for "fifth").{{Cite web|last=Pelaia|first=Ariela|title=Judaism 101: What Are the Five Books of Moses?|url=https://www.learnreligions.com/five-books-of-moses-2076335|access-date=2020-08-03|website=Learn Religions|language=en}} [405] => [406] => *The [[Hamsa|Khamsa]], an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by [[Jew]]s; that same symbol is also very popular in [[Arab]]ic culture, known to protect from envy and the [[evil eye]].{{Cite book|last=Zenner|first=Walter P.|url=https://books.google.com/books?id=xsaQeU2UFpMC&pg=PA284|title=Persistence and Flexibility: Anthropological Perspectives on the American Jewish Experience|date=1988-01-01|publisher=SUNY Press|isbn=978-0-88706-748-8|page=284|language=en}} [407] => [408] => === Christianity === [409] => *There are traditionally [[Five Wounds|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 [[Holy Lance|Spear Wound]] of Christ (respectively at the four extremities of the body, and the head).{{Cite web|title=CATHOLIC ENCYCLOPEDIA: The Five Sacred Wounds|url=https://www.newadvent.org/cathen/15714a.htm|access-date=2020-08-02|website=www.newadvent.org}} [410] => [411] => === Islam === [412] => *The [[Five Pillars of Islam]].{{Cite web|title=PBS – Islam: Empire of Faith – Faith – Five Pillars|url=https://www.pbs.org/empires/islam/faithpillars.html|access-date=2020-08-03|website=www.pbs.org}} [413] => [414] => === Gnosticism === [415] => *The number five was an important symbolic number in [[Manichaeism]], with heavenly beings, concepts, and others often grouped in sets of five. [416] => [417] => === Elements === [418] => [419] => *According to ancient Greek philosophers such as [[Aristotle]], the universe is made up of five [[classical element]]s: [[water (classical element)|water]], [[earth (classical element)|earth]], [[air (classical element)|air]], [[fire (classical element)|fire]], and [[aether (classical element)|ether]]. This concept was later adopted by medieval [[alchemists]] and more recently by practitioners of [[Neo-Pagan]] religions such as [[Wicca]]. [420] => *There are [[Pancha Bhoota|five elements]] in the universe according to [[Hindu cosmology]]: {{Transliteration|sa|dharti, agni, jal, vayu evam akash}} (earth, fire, water, air and space, respectively). [421] => *The 5 Elements of traditional Chinese ''[[Wuxing (Chinese philosophy)|Wuxing]].''{{Cite journal|last=Chen|first=Yuan|date=2014|title=Legitimation Discourse and the Theory of the Five Elements in Imperial China|url=https://muse.jhu.edu/article/611399|journal=Journal of Song-Yuan Studies|language=en|volume=44|issue=1|pages=325–364|doi=10.1353/sys.2014.0000|s2cid=147099574|issn=2154-6665}} [422] => *In [[East Asia]]n tradition, there are five elements: ([[water (Wu Xing)|water]], [[fire (Wu Xing)|fire]], [[earth (Wu Xing)|earth]], [[tree (Wu Xing)|wood]], and [[metal (Wu Xing)|metal]]).{{Cite book|last=Yoon|first=Hong-key|url=https://books.google.com/books?id=63DpqIGNjh8C&q=five+elements:+(water,+fire,+earth,+wood,+and+metal)&pg=PA59|title=The Culture of Fengshui in Korea: An Exploration of East Asian Geomancy|date=2006|publisher=Lexington Books|isbn=978-0-7391-1348-6|page=59|language=en|quote=The first category is the Five Agents [Elements] namely, Water, Fire, Wood, Metal, and Earth.}} The [[Japanese language|Japanese]] names for the [[week-day names|days of the week]], Tuesday through [[Saturday]], come from these elements via the identification of the elements with the [[Classical planet|five planets visible with the naked eye]].{{Cite book|last=Walsh|first=Len|url=https://books.google.com/books?id=QcrXBQAAQBAJ&q=The+Japanese+names+for+the+days+of+the+week,+Tuesday+through+Saturday,+come+from+these+elements&pg=PT119|title=Read Japanese Today: The Easy Way to Learn 400 Practical Kanji|date=2008-11-15|publisher=Tuttle Publishing|isbn=978-1-4629-1592-7|language=en|quote=The Japanese names of the days of the week are taken from the names of the seven basic nature symbols}} 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. [423] => [424] => [[Aether (classical element)|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.{{Cite book|last1=Kronland-Martinet|first1=Richard|url=https://books.google.com/books?id=3XFqCQAAQBAJ&q=Quintessence,+meaning+%22fifth+element%22,&pg=PA502|title=Computer Music Modeling and Retrieval. Sense of Sounds: 4th International Symposium, CMMR 2007, Copenhagen, Denmark, August 2007, Revised Papers|last2=Ystad|first2=Sølvi|last3=Jensen|first3=Kristoffer|date=2008-07-19|publisher=Springer|isbn=978-3-540-85035-9|page=502|language=en|quote=Plato and Aristotle postulated a fifth state of matter, which they called "idea" or quintessence" (from "quint" which means "fifth")}} The [[pentagram]], or five-pointed star, bears mystic significance in various belief systems including [[Baháʼí Faith|Baháʼí]], [[Christianity]], [[Freemasonry]], [[Satanism]], [[Taoism]], [[Thelema]], and [[Wicca]]. In [[numerology]], 5 or a series of [[555 (number)|555]], is often associated with change, evolution, love and abundance.{{cn|date=January 2024}} [425] => [426] => == Miscellaneous == [427] => [[File:5 playing cards.jpg|thumb|250px|The fives of all four suits in [[playing card]]s]] [428] => *"Give me five" is a common phrase used preceding a [[high five]]. [429] => *The [[Olympic Games]] have five interlocked rings as their symbol, representing the number of inhabited [[continent]]s represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).{{Cite web|date=2020-06-23|title=Olympic Rings – Symbol of the Olympic Movement|url=https://www.olympic.org/olympic-rings|access-date=2020-08-02|website=International Olympic Committee|language=en}} [430] => *The number of dots in a [[quincunx]].{{Cite book|last=Laplante|first=Philip A.|url=https://books.google.com/books?id=zoAqBgAAQBAJ&q=quincunx.five+points&pg=PA562|title=Comprehensive Dictionary of Electrical Engineering|date=2018-10-03|publisher=CRC Press|isbn=978-1-4200-3780-7|page=562|language=en|quote=quincunx five points}} [431] => [432] => == See also == [433] => {{Portal|Mathematics}} [434] => [[5 (disambiguation)]] [435] => [436] => == Notes == [437] => {{Notelist|15em}} [438] => [439] => == References == [440] => {{Reflist}} [441] => [442] => === Further reading === [443] => *{{Cite book |last=Wells |first=D. |title=The Penguin Dictionary of Curious and Interesting Numbers |title-link=The Penguin Dictionary of Curious and Interesting Numbers |location=London, UK |publisher=[[Penguin Group]] |date=1987 |pages=58–67}} [444] => [445] => == External links == [446] => *[http://primes.utm.edu/curios/page.php/5.html Prime curiosities: 5] [447] => *{{Commons category-inline}} [448] => [449] => {{Integers|zero}} [450] => {{Authority control}} [451] => [452] => {{DEFAULTSORT:5 (Number)}} [453] => [[Category:Integers]] [454] => [[Category:5 (number)]] [] => )

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