Five is the third prime number, and more specifically, the second super-prime since its prime index is prime.[1] Aside from being the sum of the only consecutive positive integers to also be prime numbers, 2 + 3, it is also the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7);[7][8] this makes it the first balanced prime with equal-sized prime gaps above and below it (of 2).[9] 5 is the first safe prime[10] where for a prime is also prime (2), and the first good prime, since it is the first prime number whose square (25) is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes (i.e., 3 × 7 = 21 and 11 × 2 = 22 are less than 25).[11] 11, the fifth prime number, is the next good prime, that also forms the first pair of sexy primes with 5.[12] More significantly, the fifth Heegner number that forms an imaginaryquadratic field with unique factorization is also 11[13] (and the first repunit primeindecimal, a base in-which five is also the first non-trivial 1-automorphic number).[14] 5 is also an Eisenstein prime (like 11) with no imaginary part and real part of the form .[1]
5 is the first prime number (and more generally, natural number) that is palindromic for a base where , with adjacent numbers 4 and 6 the only two composite numbers to be strictly non-palindromic in such sense.[15] In other words, all numbers greater than 6 in this sequence are prime, where 11 is the next strictly non-palindromic number after 6, equal to the sum of all non-prime entries in the sequence (0, 1, 4, 6). Positive integers have representations as sums of three palindromic numbers only in bases greater than or equal to five (quinary).[16]
There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime = 223058...93951 − 1 too large to compute with current computers. In a related sequence, the first five terms in the sequence of Catalan–Mersenne numbers are the only known prime terms, with a sixth possible candidate in the order of 101037.7094. These prime sequences are conjectured to be prime up to a certain limit.
The sums of the first five non-primes greater than zero 1 + 4 + 6 + 8 + 9 and the first five prime numbers 2 + 3 + 5 + 7 + 11 both equal 28; the seventh triangular number and like 6 a perfect number, which also includes 496, the thirty-first triangular number and perfect number of the form () with a of, by the Euclid–Euler theorem.[30][31][32] Within the larger family of Ore numbers, 140 and 496, respectively the fourth and sixth indexed members, both contain a set of divisors that produce integer harmonic means equal to 5.[33][34] The fifth Mersenne prime, 8191,[18] splits into 4095 and 4096, with the latter being the fifth superperfect number[35] and the sixth power of four, 46.
Five is also the total number of known unitary perfect numbers, which are numbers that are the sums of their positive properunitary divisors.[36][37] The smallest such number is 6, and the largest of these is equivalent to the sum of 4095 divisors, where 4095 is the largest of five Ramanujan–Nagell numbers that are both triangular numbers and Mersenne numbers of the general form.[38][39]
The factorial of five ismultiply perfect like 28 and 496.[40] It is the sum of the first fifteen non-zero positive integers and 15th triangular number, which in-turn is the sum of the first five non-zero positive integers and 5th triangular number. Furthermore, , where 125 is the second number to have an aliquot sum of 31 (after the fifth poweroftwo, 32).[41]
55 is the fifteenth discrete biprime,[54] equal to the product between 5 and the fifth prime and third super-prime 11.[1] These two numbers also form the second pair (5, 11) of Brown numbers such that where five is also the second number that belongs to the first pair (4, 5); altogether only five distinct numbers (4, 5, 7, 11, and 71) are needed to generate the set of known pairs of Brown numbers, where the third and largest pair is (7, 71).[55][56]
Fifty-five is also the tenth Fibonacci number,[57] whose digit sum is also 10, in its decimal representation. It is the tenth triangular number and the fourth that is doubly triangular,[58] the fifth heptagonal number[59] and fourth centered nonagonal number,[60] and as listed above, the fifth square pyramidal number.[44] The sequence of triangular that are powers of 10 is: 55, 5050, 500500, ...[61] 55 in 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,[62] with forty-five itself the ninth triangular number where 5 lies midway between 1 and 9 in the sequence of natural numbers. 45 is also conjectured by Ramsey number,[63][64] and is a Schröder–Hipparchus number; the next and fifth such number is 197, the forty-fifth prime number[17] that represents the number of ways of dissecting a heptagon into smaller polygons by inserting diagonals.[65] A five-sided convexpentagon, on the other hand, has eleven ways of being subdivided in such manner.
Five is conjectured to be the only odd, untouchable number; if this is the case, then five will be the only odd prime number that is not the base of an aliquot tree.[66]
Where five is the third prime number and odd number, every odd number greater than five is conjectured to be expressible as the sum of three prime numbers; Helfgott has provided a proof of this[67] (also known as the odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes peer-review. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).[68]
In the Collatz3x + 1problem, 5 requires five steps to reach one by multiplying terms by three and adding one if the term is odd (starting with five itself), and dividing by two if they are even: {5 ➙ 16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}; the only other number to require five steps is 32 since 16 must be part of such path (see image to the right for a map of orbits for small odd numbers).[74][75]
Specifically, 120 needs fifteen steps to arrive at 5: {120 ➙ 60 ➙ 30 ➙ 15 ➙ 46 ➙ 23 ➙ 70 ➙ 35 ➙ 106 ➙ 53 ➙ 160 ➙ 80 ➙ 40 ➙ 20 ➙ 10 ➙ 5}. These comprise a total of sixteen numbers before cycling through {16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}, where 16 is the smallest number with exactly five divisors,[76] and one of only two numbers to have an aliquot sum of 15, the other being 33.[41] Otherwise, the trajectory of 15 requires seventeen steps to reach 1,[75] where its reduced Collatz trajectory is equal to five when counting the steps {23, 53, 5, 2, 1} that are prime, including 1.[77] Overall, thirteen numbers in the Collatz map for 15 back to 1 are composite,[74] where the largest prime in the trajectory of 120 back to {4 ➙ 2 ➙ 1 ➙ 4 ➙ ...} is the sixteenth prime number, 53.[17]
When generalizing the Collatz conjecture to all positive or negative integers, −5 becomes one of only four known possible cycle starting points and endpoints, and in its case in five steps too: {−5 ➙ −14 ➙ −7 ➙ −20 ➙ −10 ➙ −5 ➙ ...}. The other possible cycles begin and end at −17 in eighteen steps, −1 in two steps, and 1 in three steps. This behavior is analogous to the path cycle of five in the 3x − 1 problem, where 5 takes five steps to return cyclically, in this instance by multiplying terms by three and subtracting 1 if the terms are odd, and also halving if even.[78] It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).[79]
In the Fibonacci sequence, which can be defined in terms of the golden ratio (see for example, Binet's formula), 5 is strictly the fifth Fibonacci number (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...) — being the sum of 2 and 3 — [1] as the only Fibonacci number greater than 1 that is equal to its position. In planar geometry, the ratio of a side and diagonal of a regular five-sided pentagon is also . Similarly, 5 is a member of the Perrin sequence, where 5 is both the fifth and sixth Perrin numbers, following (2, 3, 2) and preceding (7, 17);[80] this sequence is instead associated with the plastic ratio, the least "small" Pisot–Vijayaraghavan number that does not supersede the golden ratio.[81] This ratio is also associated with the Padovan sequence (1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, ...) where 5 is the twelfth member (and 12 the fifteenth), in-which the −th Padovan number satisfies and [82] Manipulating Narayana's cows sequence that has relations in proportion with the supergolden ratio as the fourth-smallest Pisot-Vijayaraghavan number whose value is less than the golden ratio, such that , five appears as the fourth member: (1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, ...).[83][84] On the other hand, 5 is part of the sequence of Pell numbers as the third indexed member, (0, 1, 2, 5, 12, 29, 70, 169, 408, ...).[85] These numbers are approximately proportional to powers of the second-smallest Pisot Vijayaraghavan number following , the silver ratio (and analogous to Fibonacci numbers, as powers of ), that appears in the regular octagon.
There are five countably infinite Ramsey classesofpermutations, where the age of each countable homogeneous permutation forms an individual Ramsey class ofobjects such that, for each natural number and each choice of objects , there is no object where in any -coloring of all subobjectsofisomorphicto there exists a monochromatic subobject isomorphic to .[86]: pp.1, 2 Aside from , the five classes of Ramsey permutations are the classes of:[86]: p.4
5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. Its array has a magic constantof, where the sums of its rows, columns, and diagonals are all equal to fifteen.[87] On the other hand, a normal magic square[a] has a magic constant of, where 5 and 13 are the first two Wilson primes.[26] The fifth number to return for the Mertens functionis65,[88] with counting the number of square-free integers up to with an even number of prime factors, minus the count of numbers with an odd number of prime factors. 65 is the nineteenth biprime with distinct prime factors,[54] with an aliquot sum of 19 as well[41] and equivalent to 15 + 24 + 33 + 42 + 51.[89] It is also the magic constant of the Queens Problem for ,[90] the fifth octagonal number,[91] and the Stirling number of the second kind that represents sixty-five ways of dividing a set of six objects into four non-empty subsets.[92] 13 and 5 are also the fourth and third Markov numbers, respectively, where the sixth member in this sequence (34) is the magic constant of a normal magic octagram and magic square.[93] In between these three Markov numbers is the tenth prime number 29[17] that represents the number of pentacubes when reflections are considered distinct; this number is also the fifth Lucas prime after 11 and 7 (where the first prime that is not a Lucas prime is 5, followed by 13).[94] A magic constant of 505 is generated by a normal magic square,[93] where 10 is the fifth composite.[95]
5 is also the value of the central cell the only non-trivial normal magic hexagon made of nineteen cells.[96][b] Where the sum between the magic constants of this order-3 normal magic hexagon (38) and the order-5 normal magic square (65) is 103 — the prime index of the third Wilson prime 563 equal to the sum of all three pairs of Brown numbers — their difference is 27, itself the prime index of 103.[17] In base-ten, 15 and 27 are the only two-digit numbers that are equal to the sum between their digits (inclusive, i.e. 2 + 3 + ... + 7 = 27), with these two numbers consecutive perfect totient numbers after 3 and 9.[97] 103 is the fifth irregular prime[98] that divides the numerator (236364091) of the twenty-fourth Bernoulli number, and as such it is part of the eighth irregular pair (103, 24).[99] In a two-dimensional array, the number of planar partitions with a sum of four is equal to thirteen and the number of such partitions with a sum of five is twenty-four,[100] a value equal to the sum-of-divisors of the ninth arithmetic number15[101] whose divisors also produce an integer arithmetic meanof6[102] (alongside an aliquot sum of 9).[41] The smallest value that the magic constant of a five-pointed magic pentagram can have using distinct integers is 24.[103][c]
The chromatic number of the plane is at least five, depending on the choice of set-theoretical axioms: the minimum number of colors required to color the plane such that no pair of points at a distance of 1 has the same color.[110][111] Whereas the hexagonal Golomb graph and the regular hexagonal tiling generate chromatic numbers of 4 and 7, respectively, a chromatic coloring of 5 can be attained under a more complicated graph where multiple four-coloring Moser spindles are linked so that no monochromatic triples exist in any coloring of the overall graph, as that would generate an equilateral arrangement that tends toward a purely hexagonal structure.
Space-fillingconvex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium.[118] The cube is the only Platonic solid that can tessellate space on its own, and the truncated octahedron and gyrobifastigium are the only Archimedean and Johnson solids, respectively, that can tessellate space with their own copies.
The grand antiprism, which is the only known non-Wythoffian construction of a uniform polychoron, is made of twenty pentagonal antiprisms and three hundred tetrahedra, with a total of one hundred vertices and five hundred edges.[124]
Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora based on only twenty-five uniform polyhedra: , , , , and , accompanied by a fifth or sixth general group of unique 4-prisms of Platonic and Archimedean solids. There are also a total of five Coxeter groups that generate non-prismatic Euclidean honeycombs in 4-space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite facets, as with the order-5 5-cell honeycomb and the order-5 120-cell honeycomb, both of which have five cells around each face. Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10. Likewise, analogues of four-dimensional hexadecachoricoricositetrachoric symmetry do not exist in dimensions ⩾ ; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have and symmetry. There are also five regular projective 4-polytopes in the fourth dimension, all of which are hemi-polytopes of the regular 4-polytopes, with the exception of the 5-cell.[127] Only two regular projective polytopes exist in each higher dimensional space.
There are five complex exceptional Lie algebras: , , , , and . The smallest of these, ofreal dimension 28, can be represented in five-dimensional complex space and projected as a ball rolling on top of another ball, whose motion is described in two-dimensional space.[132] is the largest, and holds the other four Lie algebras as subgroups, with a representation over in dimension 496. It contains an associated lattice that is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the 600-cell, whose Euclidean norms define a quadratic form on a lattice structure isomorphic to the optimal configuration of spheres in eight dimensions.[133] This sphere packing lattice structure in 8-space is held by the vertex arrangement of the 521 honeycomb, one of five Euclidean honeycombs that admit Gosset's original definition of a semi-regular honeycomb, which includes the three-dimensional alternated cubic honeycomb.[134][135] The smallest simple isomorphism found inside finite simple Lie groupsis,[136] where here represents alternating groups and classical Chevalley groups. In particular, the smallest non-solvable group is the alternating group on five letters, which is also the smallest simple non-abelian group.
Sporadic groups
[edit]This diagram shows the subquotient relations of the twenty-six sporadic groups; the five Mathieu groups form the simplest class (colored red ).
There are five non-supersingular prime numbers — 37, 43, 53, 61, and 67 — less than 71, which is the largest of fifteen supersingular primes that divide the order of the friendly giant, itself the largest sporadic group.[140] In particular, a centralizer of an element of order 5 inside this group arises from the product between Harada–Norton sporadic group and a group of order 5.[141][142] On its own, can be represented using standard generators that further dictate a condition where .[143][144] This condition is also held by other generators that belong to the Tits group,[145] the only finite simple group that is a non-strict group of Lie type that can also classify as sporadic (fifth-largest of all twenty-seven by order, too). Furthermore, over the field with five elements, holds a 133-dimensional representation where 5 acts on a commutative yet non-associative product as a 5-modular analogue of the Griess algebra♮,[146] which holds the friendly giant as its automorphism group.
All multiples of 5 will end in either 5 or 0, and vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions because they are prime factors of 10, the base.
In the powers of 5, every power ends with the number five, and from 53 onward, if the exponent is odd, then the hundreds digit is 1, and if it is even, the hundreds digit is 6.
A number raised to the fifth power always ends in the same digit as .
The evolution of the modern Western digit for the numeral for five is traced back to the Indian system of numerals, where on some earlier versions, the numeral bore resemblance to variations of the number four, rather than "5" (as it is represented today). The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. Later on, Arabic traditions transformed the digit in several ways, producing forms that were still similar to the numeral for four, with similarities to the numeral for three; yet, still unlike the modern five.[150] It was from those digits that Europeans finally came up with the modern 5 (represented in writings by Dürer, for example).
While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in .
On the 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.
Modern musical notation uses a musical staff made of five horizontal lines.[156] A scale with five notes per octave is called a pentatonic scale.[157]Aperfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[158]Inharmonics, the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A majortriadchord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third.
Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.
The Book of Numbers is one of five books in the Torah; the others being the books of Genesis, Exodus, Leviticus, and Deuteronomy. They are collectively called the Five Books of Moses, the Pentateuch (Greek for "five containers", referring to the scroll cases in which the books were kept), or Humash (חומש, Hebrew for "fifth").[159] The Khamsa, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by Jews; that same symbol is also very popular in Arabic culture, known to protect from envy and the evil eye.[160]
There are traditionally five woundsofJesus ChristinChristianity: the nail wounds in Christ's two hands, the nail wounds in Christ's two feet, and the Spear Wound of Christ (respectively at the four extremities of the body, and the head).[161]
According to ancient Greek philosophers such as Aristotle, the universe is made up of five classical elements: water, earth, air, fire, and ether. This concept was later adopted by medieval alchemists and more recently by practitioners of Neo-Pagan religions such as Wicca. There are five elements in the universe according to Hindu cosmology: dharti, agni, jal, vayu evam akash (earth, fire, water, air and space, respectively). In East Asian tradition, there are five elements: water, fire, earth, wood, and metal.[163] The Japanese names for the days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the five planets visible with the naked eye.[164] Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday. There are also five elements in the traditional Chinese Wuxing.[165]
Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth), as a union of these.[166] The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.
"Give me five" is a common phrase used preceding a high five.
The Olympic Games have five interlocked rings as their symbol, representing the number of inhabited continents represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).[167]
^Bourcereau (2015-08-19). "28". Prime Curios!. PrimePages. Retrieved 2022-10-13. The only known number which can be expressed as the sum of the first non-negative integers (1 + 2 + 3 + 4 + 5 + 6 + 7), the first primes (2 + 3 + 5 + 7 + 11) and the first non-primes (1 + 4 + 6 + 8 + 9). There is probably no other number with this property.
Only twelve integers up to 33 cannot be expressed as the sum of five non-zero squares: {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33} where 2, 3 and 7 are the only such primes without an expression.
^Weisstein, Eric W. "Perrin Sequence". mathworld.wolfram.com. Retrieved 2020-07-30.
^M.J. Bertin; A. Decomps-Guilloux; M. Grandet-Hugot; M. Pathiaux-Delefosse; J.P. Schreiber (1992). Pisot and Salem Numbers. Birkhäuser. ISBN3-7643-2648-4.
^Alon, Noga; Grytczuk, Jaroslaw; Hałuszczak, Mariusz; Riordan, Oliver (2002). "Nonrepetitive colorings of graphs"(PDF). Random Structures & Algorithms. 2 (3–4): 337. doi:10.1002/rsa.10057. MR1945373. S2CID5724512. A coloring of the set of edges of a graph G is called non-repetitive if the sequence of colors on any path in G is non-repetitive...In Fig. 1 we show a non-repetitive 5-coloring of the edges of P... Since, as can easily be checked, 4 colors do not suffice for this task, we have π(P) = 5.
^Wills, J. M. (1987). "The combinatorially regular polyhedra of index 2". Aequationes Mathematicae. 34 (2–3): 206–220. doi:10.1007/BF01830672. S2CID121281276.
"In tables 4 to 8, we list the seventy-five nondihedral uniform polyhedra, as well as the five pentagonal prisms and antiprisms, grouped by generating Schwarz triangles." Appendix II: Uniform Polyhedra.
^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
^Kisia, S. M. (2010), Vertebrates: Structures and Functions, Biological Systems in Vertebrates, CRC Press, p. 106, ISBN978-1-4398-4052-8, The typical limb of tetrapods is the pentadactyl limb (Gr. penta, five) that has five toes. Tetrapods evolved from an ancestor that had limbs with five toes. ... Even though the number of digits in different vertebrates may vary from five, vertebrates develop from an embryonic five-digit stage.
^Cinalli, G.; Maixner, W. J.; Sainte-Rose, C. (2012-12-06). Pediatric Hydrocephalus. Springer Science & Business Media. p. 19. ISBN978-88-470-2121-1. The five appendages of the starfish are thought to be homologous to five human buds
^Veith (Jr.), Gene Edward; Wilson, Douglas (2009). Omnibus IV: The Ancient World. Veritas Press. p. 52. ISBN978-1-932168-86-0. The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter)