5
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Cardinal | five | |||
Ordinal | 5th (fifth) | |||
Tamil | ௫ | |||
Thai | ๕ | |||
Babylonian numeral | 𒐙 | |||
Egyptian hieroglyph, Chinese counting rod | ||||| | |||
Maya numerals | 𝋥 | |||
Morse code | ..... |
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
Evolution of the Arabic digit
The evolution of the modern Western digit for the numeral 5 cannot be traced back to the Indian system, as opposed to digits 1 to 4. The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. The Nagari and Punjabi took these digits and all came up with forms that were similar to a lowercase "h" rotated 180°. The Ghubar Arabs transformed the digit in several ways, producing from that were more similar to the digits 4 or 3 than to 5.[1] It was from those digits that Europeans finally came up with the modern 5.
While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in .
On the seven-segment display of a calculator and digital clock, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa. It is one of three numbers, along with 4 and 6, where the number of segments matches the number.
Mathematics
Five is the third-smallest
part of the form .[2] In particular, five is the first congruent number, since it is the length of the hypotenuse of the smallest integer-sided right triangle.[17]Number theory
5 is the fifth
5 is the second Fermat prime of the form , and more generally the second Sierpiński number of the first kind, .
5 is also the third Mersenne prime exponent of the form , which yields , the eleventh prime number and fifth
There are a total of five known
Figurate numbers
In
- 5 is a indexed member in the sequence is 55.
- 5 is a centered square number: 1, 5, 13, 25, 41, ...[40] The fifth square number or 52 is 25, which features in the proportions of the two smallest (3, 4, 5) and (5, 12, 13) primitive Pythagorean triples.[41]
The factorial of five is multiply perfect like 28 and 496.[42] It is the sum of the first fifteen non-zero positive integers and 15th triangular number, which in-turn is the sum of the first five non-zero positive integers and 5th triangular number. Furthermore, , where 125 is the second number to have an aliquot sum of 31 (after the fifth power of two, 32).[43] On its own, 31 is the first prime centered pentagonal number,[44] and the fifth centered triangular number.[45] Collectively, five and thirty-one generate a sum of 36 (the square of 6) and a difference of 26, which is the only number to lie between a square and a cube (respectively, 25 and
Magic figures
5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. Its array has a magic constant of , where the sums of its rows, columns, and diagonals are all equal to fifteen.[64] On the other hand, a normal magic square[b] has a magic constant of , where 5 and 13 are the first two Wilson primes.[4] The fifth number to return for the Mertens function is 65,[65] with counting the number of square-free integers up to with an even number of
5 is also the value of the central cell the only non-trivial
Collatz conjecture
In the Collatz 3x + 1 problem, 5 requires five steps to reach one by multiplying terms by three and adding one if the term is odd (starting with five itself), and dividing by two if they are even: {5 ➙ 16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}; the only other number to require five steps is 32 since 16 must be part of such path (see [e] for a map of orbits for small odd numbers).[81][82]
Specifically, 120 needs fifteen steps to arrive at 5: {120 ➙ 60 ➙ 30 ➙ 15 ➙ 46 ➙ 23 ➙ 70 ➙ 35 ➙ 106 ➙ 53 ➙ 160 ➙ 80 ➙ 40 ➙ 20 ➙ 10 ➙ 5}. These comprise a total of sixteen numbers before cycling through {16 ➙ 8 ➙ 4 ➙ 2 ➙ 1}, where 16 is the smallest number with exactly five divisors,[83] and one of only two numbers to have an aliquot sum of 15, the other being 33.[43] Otherwise, the trajectory of 15 requires seventeen steps to reach 1,[82] where its reduced Collatz trajectory is equal to five when counting the steps {23, 53, 5, 2, 1} that are prime, including 1.[84] Overall, thirteen numbers in the Collatz map for 15 back to 1 are composite,[81] where the largest prime in the trajectory of 120 back to {4 ➙ 2 ➙ 1 ➙ 4 ➙ ...} is the sixteenth prime number, 53.[24]
When generalizing the Collatz conjecture to all positive or negative integers, −5 becomes one of only four known possible cycle starting points and endpoints, and in its case in five steps too: {−5 ➙ −14 ➙ −7 ➙ −20 ➙ −10 ➙ −5 ➙ ...}. The other possible cycles begin and end at −17 in eighteen steps, −1 in two steps, and 1 in three steps. This behavior is analogous to the path cycle of five in the 3x − 1 problem, where 5 takes five steps to return cyclically, in this instance by multiplying terms by three and subtracting 1 if the terms are odd, and also halving if even.[85] It is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).[86]
Generalizations
Is 5 the only odd untouchable number?
Five is conjectured to be the only odd untouchable number, and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.[87] Meanwhile:
- Every odd number greater than is the sum of at most five prime numbers,[88] and
- Every odd number greater than is conjectured to be expressible as the sum of three prime numbers. peer-review.
As a consequence of
can be expressed as the sum of five non-zero squares.[91][92]Regarding Waring's problem, , where every natural number is the sum of at most thirty-seven fifth powers.[93][94]
There are five countably infinite Ramsey classes of permutations, where the age of each countable homogeneous permutation forms an individual Ramsey class of objects such that, for each natural number and each choice of objects , there is no object where in any -coloring of all subobjects of isomorphic to there exists a monochromatic subobject isomorphic to .[95]: pp.1, 2 Aside from , the five classes of Ramsey permutations are the classes of:[95]: p.4
- Identity permutations, and reversals
- Increasing sequences of decreasing sequences, and decreasing sequences of increasing sequences
- All permutations
In general, the Fraïssé limit of a class of finite relational structure is the age of a countable homogeneous relational structure if and only if five conditions hold for : it is closed under isomorphism, it has only countably many isomorphism classes, it is hereditary, it is joint-embedded, and it holds the amalgamation property.[95]: p.3
In the general
Geometry
A
Graphs theory, and planar geometry
In
ofThe
The plane also contains a total of five
Polyhedra
There are five
- stella octangula.[109] Icosahedral symmetryisisomorphic to the alternating groupon five letters of ordermirror planesof pass through the edges of a regularhyperbolic plane.
Moreover, the fifth
Fourth dimension
The pentatope, or 5-cell, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry of order 120 = 5! and
: p.120- A duoantiprismatic solution in the fourth dimension, constructed from the polytope cartesian productand made of fiftypentagrammic crossed antiprisms, ten pentagonal antiprisms, and fifty vertices.[117]: p.124.
- The grand antiprism, which is the only known non-Wythoffian construction of a uniform polychoron, is made of twenty pentagonal antiprisms and three hundred tetrahedra, with a total of one hundred vertices and five hundred edges.[118]
- The abstract four-dimensional 57-cell is made of fifty-seven hemi-icosahedral cells, in-which five surround each edge.[119] The 11-cell, another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven hemi-dodecahedral cells each with fifteen edges.[120] The skeleton of the hemi-dodecahedron is the Petersen graph
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
In particular, Bring's surface is the curve in the projective plane that is represented by the
It holds the largest possible
Fifth dimension
The 5-simplex or hexateron is the five-dimensional analogue of the 5-cell, or 4-simplex. It has Coxeter group as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group , the only finite symmetric group which has an
A Veronese surface in the projective plane generalizes a linear condition for a point to be contained inside a
Finite simple groups
There are five complex exceptional Lie algebras: , , , , and . The smallest of these, of
is the largest, and holds the other four Lie algebras as subgroups, with a representation over in dimension 496. It contains an associatedThe five
There are five non-supersingular prime numbers — 37, 43, 53, 61, and 67 — less than 71, which is the largest of fifteen supersingular primes that divide the order of the friendly giant, itself the largest sporadic group.[134] In particular, a centralizer of an element of order 5 inside this group arises from the product between Harada–Norton sporadic group and a group of order 5.[135][136] On its own, can be represented using standard generators that further dictate a condition where .[137][138] This condition is also held by other generators that belong to the Tits group ,
Euler's identity
Euler's identity, + = , contains five essential
List of basic calculations
Multiplication | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5 × x | 5 | 10
|
15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 | 95 | 100
|
Division | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5 ÷ x | 5 | 2.5 | 1.6 | 1.25 | 1 | 0.83 | 0.714285 | 0.625 | 0.5 | 0.5 | 0.45 | 0.416 | 0.384615 | 0.3571428 | 0.3 | |
x ÷ 5 | 0.2 | 0.4 | 0.6 | 0.8 | 1.2 | 1.4 | 1.6 | 1.8 | 2 | 2.2 | 2.4 | 2.6 | 2.8 | 3 |
Exponentiation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5x | 5 | 25 | 125 | 625
|
3125 | 15625 | 78125 | 390625 | 1953125 | 9765625 | 48828125 | 244140625 | 1220703125 | 6103515625 | 30517578125 | |
x5 | 1 | 32 | 243 | 1024 | 7776 | 16807 | 32768 | 59049 | 100000 | 161051 | 248832 | 371293 | 537824 | 759375 |
In decimal
All multiples of 5 will end in either 5 or
, the base.In the
A number raised to the fifth power always ends in the same digit as .
Science
Astronomy
- There are five Lagrangian pointsin a two-body system.
Biology
- There are usually considered to be
- Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.[145]
- Five is the number of appendages on most pentamerism.[146]
Computing
- 5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.[147]
Literature
Poetry
A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.[148]
Music
- Modern musical notation uses a musical staff made of five horizontal lines.[149]
- A scale with five notes per octave is called a pentatonic scale.[150]
- A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[151]
- In harmonics, the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A major triad chord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third.
- Five is the lowest possible number that can be the top number of a meter.
Religion and mysticism
Judaism
- 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").[152]
- The
Christianity
- There are traditionally Jesus Christ in Christianity: the nail wounds in Christ's two hands, the nail wounds in Christ's two feet, and the Spear Wound of Christ (respectively at the four extremities of the body, and the head).[154]
Islam
Gnosticism
- The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.
Elements
- According to ancient Greek philosophers such as 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).
- The 5 Elements of traditional Chinese Wuxing.[156]
- In days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the five planets visible with the naked eye.[158]Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday.
Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth), as a union of these.[159] The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca. In numerology, 5 or a series of 555, is often associated with change, evolution, love and abundance.[citation needed]
Miscellaneous
- "Give me five" is a common phrase used preceding a high five.
- The Olympic Games have five interlocked rings as their symbol, representing the number of inhabited continents represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).[160]
- The number of dots in a quincunx.[161]
See also
Notes
References
- Harvill Press(1998): 394, Fig. 24.65
- ^ a b c d e f g Weisstein, Eric W. "5". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes p: (p-1)/2 is also prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A007540 (Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A006562 (Balanced primes (of order one): primes which are the average of the previous prime and the following prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
- ^ Sloane, N. J. A. (ed.). "Sequence A080076 (Proth primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
- ^ Weisstein, Eric W. "Mersenne Prime". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ Weisstein, Eric W. "Catalan Number". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-20.
- ^ Sloane, N. J. A. (ed.). "Sequence A003226 (Automorphic numbers: m^2 ends with m.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A088054 (Factorial primes: primes which are within 1 of a factorial number.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
- ^ Weisstein, Eric W. "Twin Primes". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
- ^ Weisstein, Eric W. "Perrin Sequence". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ Weisstein, Eric W. "Sierpiński Number of the First Kind". mathworld.wolfram.com. Retrieved 2020-07-30.
- ^ Sloane, N. J. A. (ed.). "Sequence A019434 (Fermat primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21.
- ^ Sloane, N. J. A. (ed.). "Sequence A004729 (... the 31 regular polygons with an odd number of sides constructible with ruler and compass)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26.
- ^ S2CID 115239655.
- ^ Sloane, N. J. A. (ed.). "Sequence 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-31.
- ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-08.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A000668 (Mersenne primes (primes of the form 2^n - 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-03.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A103901 (Mersenne primes p such that M(p) equal to 2^p - 1 is also a (Mersenne) prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-03.
- ISBN 0-387-20860-7.
- ^ Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
- ^ Sloane, N. J. A. (ed.). "Sequence A076046 (Ramanujan-Nagell numbers: the triangular numbers...which are also of the form 2^b - 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
- ^ Sloane, N. J. A. (ed.). "Sequence A000225 (... (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-13.
- ^ 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.
- ^ Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers: numbers n such that the harmonic mean of the divisors of n is an integer.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A001600 (Harmonic means of divisors of harmonic numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A019279 (Superperfect numbers: numbers k such that sigma(sigma(k)) equals 2*k where sigma is the sum-of-divisors function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
- ^ Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
- ^ Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers...Sum of two squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
- ^ Sloane, N. J. A. (ed.). "Sequence 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-05-26.
- ^ Sloane, N. J. A. (ed.). "Sequence A007691 (Multiply-perfect numbers: n divides sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-28.
- ^ a b c d Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-11.
- ^ Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
- ^ Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-21.
- S2CID 5216897.
- ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08. In general, the sum of n consecutive triangular numbers is the nth tetrahedral number.
- ^ Sloane, N. J. A. (ed.). "Sequence 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}...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-28.
- ^ Sloane, N. J. A. (ed.). "Sequence A118372 (S-perfect numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-28.
- OCLC 317778112.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A006881 (Squarefree semiprimes: Numbers that are the product of two distinct primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A216071 (Brocard's problem: positive integers m such that m^2 equal to n! + 1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-09.
- ^ Sloane, N. J. A. (ed.). "Sequence A085692 (Brocard's problem: squares which can be written as n!+1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-09.
- ^ Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci numbers: F(n) is F(n-1) + F(n-2) with F(0) equal to 0 and F(1) equal to 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A002817 (Doubly triangular numbers: a(n) equal to n*(n+1)*(n^2+n+2)/8.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A000566 (Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A060544 (Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1) equal to 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A037156 (a(n) equal to 10^n*(10^n+1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 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
... - ^ Sloane, N. J. A. (ed.). "Sequence A006886 (Kaprekar numbers...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
- ^ Sloane, N. J. A. (ed.). "Sequence A120414 (Conjectured Ramsey number R(n,n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
- ^ Sloane, N. J. A. (ed.). "Sequence A212954 (Triangle read by rows: two color Ramsey numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
- ^ Sloane, N. J. A. (ed.). "Sequence A001003 (Schroeder's second problem; ... also called super-Catalan numbers or little Schroeder numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
- ^ William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 2022-07-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A028442 (Numbers k such that Mertens's function M(k) (A002321) is zero.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A003101 (a(n) as Sum_{k equal to 1..n} (n - k + 1)^k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ Sloane, N. J. A. (ed.). "Sequence A000567 (Octagonal numbers: n*(3*n-2). Also called star numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
- ^ "Sloane's A008277 :Triangle of Stirling numbers of the second kind". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-12-24.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
- ^ Sloane, N. J. A. (ed.). "Sequence A000162 (Number of 3-dimensional polyominoes (or polycubes) with n cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
- ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-25.
- ^ Trigg, C. W. (February 1964). "A Unique Magic Hexagon". Recreational Mathematics Magazine. Retrieved 2022-07-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-10.
- ^ Sloane, N. J. A. (ed.). "Sequence 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
- ^ Sloane, N. J. A. (ed.). "Sequence A189683 (Irregular pairs (p,2k) ordered by increasing k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-07.
- ^ Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-10.
- ^ Sloane, N. J. A. (ed.). "Sequence A000203 (a(n) is sigma(n), the sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-14.
- Zbl 0684.00001.
- ^ a b Sloane, N. J. A. (ed.). "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-01-24.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-24.
- "Table of n, a(n) for n = 1..10000"
- ^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-06.
- ^ Sloane, N. J. A. (ed.). "Sequence 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-18.
- ^ Sloane, N. J. A. (ed.). "Sequence A003079 (One of the basic cycles in the x->3x-1 (x odd) or x/2 (x even) problem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-24.
- {5 ➙ 14 ➙ 7 ➙ 20 ➙ 10 ➙ 5 ➙ ...}.
- ^ Sloane, N. J. A. (ed.). "3x-1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-01-24.
- S2CID 30344483.
- S2CID 2618958.
- OCLC 913564239.
- Zbl 1290.05015.
- ISBN 978-0-19-853171-5.
- ^ Sloane, N. J. A. (ed.). "Sequence A047701 (All positive numbers that are not the sum of 5 nonzero squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-20.
- 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.
- ^ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem: g(n) equal to 2^n + floor((3/2)^n) - 2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-20.
- S2CID 179177986.
- ^ Zbl 1267.05284.
- S2CID 60314285.
- Zbl 1032.17003.
- S2CID 124558613.
- S2CID 118322641.
- .
- ISBN 0-521-43594-3.
- S2CID 5724512.
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.
- ^ Royle, G. "Cubic Symmetric Graphs (The Foster Census)." Archived 2008-07-20 at the Wayback Machine
- S2CID 119273214.
- Zbl 1445.05040.
- Zbl 0385.51006.
- MR 0857454. Section 9.3: "Other Monohedral tilings by convex polygons".
- ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61
- ^ S2CID 123279687.
- S2CID 202679388.
- ^ Hart, George W. "Symmetry Planes". Virtual Polyhedra (The Encyclopedia of Polyhedra). Retrieved 2023-09-27.
- "They can be colored as five sets of three mutually orthogonal planes" where the "fifteen planes divide the sphere into 120 Möbius triangles."
- ISBN 978-1-58988-285-0.
- Alexandrov, A. D. (2005). "8.1 Parallelohedra". Convex Polyhedra. Springer. pp. 349–359.
- ^ Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 2022-11-26. Retrieved 2023-01-12.
- S2CID 121281276.
- Zbl 0784.51020.
- "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.
- "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."
- ^ ISBN 978-0-486-61480-9.
- ISBN 978-1-56881-220-5. Chapter 26: "The Grand Antiprism"
- S2CID 120672023..
- ISBN 978-0-444-86571-7.
- S2CID 115688843.
- Zbl 0397.51013.
- CiteSeerX 10.1.1.361.251.
- Zbl 0732.51002.
- S2CID 125356690.
- S2CID 50818244.
- Zbl 1476.51020.
- Zbl 0898.52004.
- JFM 30.0494.02.
- Zbl 0568.20001.
- ^ Zbl 0908.20007.
- Zbl 1089.20006.
- S2CID 115302359.
- S2CID 16584404.
- Zbl 1135.20007.
- Zbl 0636.20014.
- Zbl 0914.20016.
- Zbl 1087.20025.
- ^ Wilson, R.A.; Parker, R.A.; Nickerson, S.J.; Bray, J.N. (1999). "Exceptional group 2F4(2)', Tits group T". ATLAS of Finite Group Representations.
- Zbl 0851.20034.
- OCLC 990970269.
- OCLC 26361981.
- ^ Gallagher, James (13 February 2014). "Mathematics: Why the brain sees maths as beauty". BBC News Online. British Broadcasting Corporation (BBC). Retrieved 2023-06-02.
- ISBN 978-0-12-391883-3.
There are five basic tastes: sweet, salty, sour, bitter and umami...
- ISBN 978-1-4398-4052-8,
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.
- ISBN 978-88-470-2121-1.
The five appendages of the starfish are thought to be homologous to five human buds
- ISBN 978-1-4665-1228-3.
5 5 005 ENQ (enquiry)
- ISBN 978-1-932168-86-0.
The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter)
- ^ "STAVE | meaning in the Cambridge English Dictionary". dictionary.cambridge.org. Retrieved 2020-08-02.
the five lines and four spaces between them on which musical notes are written
- ISBN 978-1-4574-9410-9.
Pentatonic scales, as used in jazz, are five note scales
- ^ Danneley, John Feltham (1825). 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. editor, and pub.
are the perfect fourth, perfect fifth, and the octave
- ^ Pelaia, Ariela. "Judaism 101: What Are the Five Books of Moses?". Learn Religions. Retrieved 2020-08-03.
- ISBN 978-0-88706-748-8.
- ^ "CATHOLIC ENCYCLOPEDIA: The Five Sacred Wounds". www.newadvent.org. Retrieved 2020-08-02.
- ^ "PBS – Islam: Empire of Faith – Faith – Five Pillars". www.pbs.org. Retrieved 2020-08-03.
- S2CID 147099574.
- ISBN 978-0-7391-1348-6.
The first category is the Five Agents [Elements] namely, Water, Fire, Wood, Metal, and Earth.
- ISBN 978-1-4629-1592-7.
The Japanese names of the days of the week are taken from the names of the seven basic nature symbols
- ISBN 978-3-540-85035-9.
Plato and Aristotle postulated a fifth state of matter, which they called "idea" or quintessence" (from "quint" which means "fifth")
- ^ "Olympic Rings – Symbol of the Olympic Movement". International Olympic Committee. 2020-06-23. Retrieved 2020-08-02.
- ISBN 978-1-4200-3780-7.
quincunx five points
Further reading
- Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. London, UK: Penguin Group. pp. 58–67.
External links
- Prime curiosities: 5
- Media related to 5 (number) at Wikimedia Commons