← 4 5 6 →
Cardinalfive
Ordinal5th
(fifth)
Numeral systemquinary
Factorizationprime
Prime3rd
Divisors1,5
Greek numeralΕ´
Roman numeralV, v
Greek prefixpenta-/pent-
Latin prefixquinque-/quinqu-/quint-
Binary1012
Ternary123
Senary56
Octal58
Duodecimal512
Greekε (or Ε)
Arabic, Kurdish٥
Persian, Sindhi, Urdu۵
Ge'ez
Bengali
Punjabi
Chinese numeral五,伍
Devanāgarī
Hebrewה
Khmer
Telugu
Malayalam
Tamil
Thai

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 attained significance throughout history in part because typical humans have five digits on each hand.

In mathematics

The first Pythagorean triple.

${\displaystyle 5}$ is the third smallest prime number, and the second super-prime.[1] It is the first safe prime, the first good prime, and the first of three known Wilson primes.[2] Five is the second Fermat prime[1] and the third Mersenne prime exponent,[3] as well as the third Catalan number,[4] and the third Sophie Germain prime.[1] Notably, 5 is equal to the sum of the only consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7). It is also a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial,[5] and an Eisenstein prime with no imaginary part and real part of the form ${\displaystyle 3p}$${\displaystyle 1}$.[1] In particular, five is the first congruent number, since it is the length of the hypotenuse of the smallest integer-sided right triangle.[6]

Five is the second Fermat prime of the form ${\displaystyle 2^{2^{n))}$+ ${\displaystyle 1}$, and more generally the second Sierpiński number of the first kind, ${\displaystyle n^{n))$+ ${\displaystyle 1}$.[7] There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.[8] The sum of the first 3 Fermat primes, 3, 5 and 17, yields 25 or 52, while 257 is the 55th prime number. Combinations from these 5 Fermat primes generate 31 polygons with an odd number of sides that can be construncted purely with a compass and straight-edge, which includes the five-sided regular pentagon. Apropos, 31 is also equal to the sum of the maximum number of areas inside a circle that are formed from the sides and diagonals of the first five ${\displaystyle n}$-sided polygons, and equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem.[9]

The number 5 is the fifth Fibonacci number, being 2 plus 3.[1] It is the only Fibonacci number that is equal to its position aside from 1, which is both the first and second Fibonacci numbers. Five is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... ( lists Markov numbers that appear in solutions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers.[10]

5 is the third Mersenne prime exponent of the form ${\displaystyle 2^{n))$${\displaystyle 1}$, which yields ${\displaystyle 31}$: the prime index of the third Mersenne prime and second double Mersenne prime 127, as well as the third double Mersenne prime exponent for the number 2,147,483,647, which is the largest value that a signed 32-bit integer field can hold. There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime ${\displaystyle M_{M_{61))}$ = 223058...93951 − 1 too large to compute with current computers. In a related sequence, the first 5 terms in the sequence of Catalan–Mersenne numbers ${\displaystyle M_{c_{n))}$ 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.

Every odd number greater than ${\displaystyle 1}$ is the sum of at most five prime numbers, and every odd number greater than ${\displaystyle 5}$ can be expressed as the sum of three prime numbers.[11][12] The proof of the latter, also known as the odd Goldbach conjecture, is already widely acknowledged by mathematicians, even though it is still undergoing peer-review.

The sums of the first five non-primes greater than zero ${\displaystyle 1}$ + ${\displaystyle 4}$ + ${\displaystyle 6}$ + ${\displaystyle 8}$ + ${\displaystyle 9}$ and the first five prime numbers ${\displaystyle 2}$ + ${\displaystyle 3}$ + ${\displaystyle 5}$ + ${\displaystyle 7}$ + ${\displaystyle 11}$ both equal ${\displaystyle 28}$; the 7th triangular number and like ${\displaystyle 6}$ a perfect number, which also includes ${\displaystyle 496}$, the 31st triangular number and perfect number of the form ${\displaystyle 2^{p))$−1(${\displaystyle 2^{p))$${\displaystyle 1}$) with a ${\displaystyle p}$ of ${\displaystyle 5}$, by the Euclid–Euler theorem.[13][14][15]

There are a total of five known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors. A sixth unitary number, if discovered, would have at least nine odd prime factors.[16]

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.[17]

In figurate numbers, 5 is a pentagonal number, with the sequence of pentagonal numbers starting: 1, 5, 12, 22, 35, ...[18]

The factorial of five, or ${\displaystyle 5}$! = ${\displaystyle 120}$, 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. 35, which is the fourth or fifth pentagonal and tetrahedral number, is equal to the sum of the first five triangular numbers: 1, 3, 6, 10, 15.[22]

5 is the value of the central cell of the only non-trivial normal magic square, also called the Lo Shu square. Its ${\displaystyle 3}$ x ${\displaystyle 3}$ array of squares has a magic constant ${\displaystyle M}$ of ${\displaystyle 15}$, where the sums of its rows, columns, and diagonals are all equal to fifteen.[23] 5 is also the value of the central cell the only non-trivial order-3 normal magic hexagon that is made of nineteen cells.[24]

Polynomial equations of degree 4 and below can be solved with radicals, while quintic equations of degree 5, and higher, cannot generally be so solved. This is the Abel–Ruffini theorem. This is related to the fact that the symmetric group ${\displaystyle \mathrm {S} _{n))$ is a solvable group for n ⩽ 4 and not solvable for n ⩾ 5.

Euler's identity, ${\displaystyle e^{i\pi ))$+ ${\displaystyle 1}$ = ${\displaystyle 0}$, contains five essential numbers used widely in mathematics: Archimedes' constant ${\displaystyle \pi }$, Euler's number ${\displaystyle e}$, the imaginary number ${\displaystyle i}$, unity ${\displaystyle 1}$, and zero ${\displaystyle 0}$, which makes this formula a renown example of beauty in mathematics.

In geometry

A pentagram, or five-pointed polygram, is the first proper star polygon constructed from the diagonals of a regular pentagon as self-intersecting edges that are proportioned in golden ratio, ${\displaystyle \varphi }$. Its internal geometry appears prominently in Penrose tilings, and is a facet inside Kepler-Poinsot star polyhedra and Schläfli–Hess star polychora, represented by its Schläfli symbol {5/2}. A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges. Generally, star polytopes that are regular only exist in dimensions 2 ⩽ ${\displaystyle n}$ < 5.

In graph theory, all graphs with 4 or fewer vertices are planar, however, there is a graph with 5 vertices that is not: K5, the complete graph with 5 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 K5, or the complete bipartite utility graph K3,3.[25] A similar graph is the Petersen graph, which is strongly connected and also nonplanar. It is most easily described as graph of a pentagram embedded inside a pentagon, with a total of 5 crossings, a girth of 5, and a Thue number of 5.[26][27] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.[28] The automorphism group of the Petersen graph is the symmetric group ${\displaystyle \mathrm {S} _{5))$ of order 120 = 5!.

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.[29] 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.

The plane contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations: hexagonal, oblique, rectangular, centered rectangular, and square lattices. The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.[30]

Five points are needed to determine a conic section, in the same way that two points are needed to determine a line.[31] A Veronese surface in the projective plane ${\displaystyle \mathbb {P} ^{5))$ of a conic generalizes a linear condition for a point to be contained inside a conic.

There are ${\displaystyle 5}$ Platonic solids in three-dimensional space: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.[32] The dodecahedron in particular contains pentagonal faces, while the icosahedron, its dual polyhedron, has a vertex figure that is a regular pentagon. There are also ${\displaystyle 5}$:

Regular polyhedron compounds: the stella octangula, compound of five tetrahedra, compound of five cubes, compound of five octahedra, and compound of ten tetrahedra.[33] Icosahedral symmetry ${\displaystyle \mathrm {I} _{h))$ is isomorphic to the alternating group on 5 letters ${\displaystyle \mathrm {A} _{5))$ of order 120, realized by actions on these uniform polyhedron compounds.

Space-filling convex polyhedra: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium.[34] While the cube is the only Platonic solid that can tessellate space on its own, the truncated octahedron and the gyrobifastigium are the only Archimedean and Johnson solids, respectively, that can also tessellate space with their own copies.

Cell-transitive parallelohedra: any parallelepiped, as well as the rhombic dodecahedron and elongated dodecahedron, and the hexagonal prism and truncated octahedron.[35] The cube is a special case of a parallelepiped, with the rhombic dodecahedron the only Catalan solid to tessellate space on its own.

Regular abstract polyhedra, which include the excavated dodecahedron and the dodecadodecahedron.[36] They have combinatorial symmetries transitive on flags of their elements, with topologies equivalent to that of toroids and the ability to tile the hyperbolic plane.

The 5-cell, or pentatope, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry ${\displaystyle \mathrm {A} _{4))$ of order 120 = 5! and ${\displaystyle \mathrm {S} _{5))$ group structure. Made of five tetrahedra, its Petrie polygon is a regular pentagon and its orthogonal projection is equivalent to the complete graph K5. It is one of six regular 4-polytopes, made of thirty-one elements: five vertices, ten edges, ten faces, five tetrahedral cells and one 4-face.[37]

• 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.[39]

Overall, the fourth dimension contains five Weyl groups that form a finite number of uniform polychora: ${\displaystyle \mathrm {A} _{4))$, ${\displaystyle \mathrm {B} _{4))$, ${\displaystyle \mathrm {D} _{4))$, ${\displaystyle \mathrm {F} _{4))$, and ${\displaystyle \mathrm {H} _{4))$, with four of these Coxeter groups capable of generating the same finite forms without ${\displaystyle \mathrm {D} _{4))$; 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 Eucledian 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 three-dimensional icosahedral symmetry ${\displaystyle \mathrm {I} _{h))$ or four-dimensional ${\displaystyle \mathrm {H} _{4))$ symmetry do not exist in dimensions n ⩾ 5; however, there is the uniform prismatic group ${\displaystyle \mathrm {H} _{4))$ × ${\displaystyle \mathrm {A} _{1))$ in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have ${\displaystyle \mathrm {H} _{4))$ symmetry.

The 5-simplex is the five-dimensional analogue of the 5-cell, or 4-simplex; the fifth iteration of ${\displaystyle n}$-simplexes in any ${\displaystyle n}$ dimensions. The 5-simplex has the Coxeter group ${\displaystyle \mathrm {A} _{5))$ as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group ${\displaystyle \mathrm {S} _{6))$, the only finite symmetric group which has an outer automorphism. The 5-cube, made of ten tesseracts and the 5-cell as its vertex figure, is also regular and one of thirty-one uniform 5-polytopes under the Coxeter ${\displaystyle \mathrm {B} _{5))$ hypercubic group. The demipenteract, with one hundred and twenty cells, is the only fifth-dimensional semiregular polytope, and has the rectified 5-cell as its vertex figure, which is one of only three semiregular 4-polytopes alongside the rectified 600-cell and the snub 24-cell. In the fifth dimension, there are five regular paracompact honeycombs, all with infinite facets and vertex figures.[42] There are exclusively twelve complex aperiotopes in ${\displaystyle \mathbb {C} ^{n))$ complex spaces of dimensions ${\displaystyle n}$${\displaystyle 5}$, with fifteen in ${\displaystyle \mathbb {C} ^{4))$ and sixteen in ${\displaystyle \mathbb {C} ^{3))$; alongside complex polytopes in ${\displaystyle \mathbb {C} ^{5))$ and higher under simplex, hypercubic and orthoplex groups, the latter with van Oss polytopes.

There are five exceptional Lie groups: ${\displaystyle {\mathfrak {g))_{2))$, ${\displaystyle {\mathfrak {f))_{4))$, ${\displaystyle {\mathfrak {e))_{6))$, ${\displaystyle {\mathfrak {e))_{7))$, and ${\displaystyle {\mathfrak {e))_{8))$. The smallest of these, ${\displaystyle {\mathfrak {g))_{2))$, can be represented in five-dimensional complex space and projected in the same number of dimensions as a ball rolling on top of another ball, whose motion is described in two-dimensional space.[43] ${\displaystyle {\mathfrak {e))_{8))$, the largest of all five exceptional groups, also contains the other four as subgroups and is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the 600-cell. There are also five solvable groups that are excluded from finite simple groups of Lie type.

The five Mathieu groups constitute the first generation in the happy family of sporadic groups. These are also the first five sporadic groups to have been described, defined as ${\displaystyle \mathrm {M} _{n))$ multiply transitive permutation groups on ${\displaystyle n}$ objects, with ${\displaystyle n}$ {11, 12, 22, 23, 24}. In particular, ${\displaystyle \mathrm {M} _{11))$, the smallest of all sporadic groups, has a rank 3 action on fifty-five points from an induced action on unordered pairs, as well as two five-dimensional faithful complex irreducible representations over the field with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with n elements.[44] Of precisely five different conjugacy classes of maximal subgroups of ${\displaystyle \mathrm {M} _{11))$, one is the almost simple symmetric group ${\displaystyle \mathrm {S} _{5))$ (of order 5!), and another is ${\displaystyle \mathrm {M} _{10))$, also almost simple, that functions as a point stabilizer which has ${\displaystyle 5}$ as its largest prime factor in its group order: 24·32·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas ${\displaystyle \mathrm {M} _{11))$ is sharply 4-transitive, ${\displaystyle \mathrm {M} _{12))$ is sharply 5-transitive and ${\displaystyle \mathrm {M} _{24))$ is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups. ${\displaystyle \mathrm {M} _{22))$ has the first five prime numbers as its distinct prime factors in its order of 27·32·5·7·11, and is the smallest of five sporadic groups with five distinct prime factors in their order. All Mathieu groups are subgroups of ${\displaystyle \mathrm {M} _{24))$, which under the Witt design ${\displaystyle \mathrm {W} _{24))$ of Steiner system S(5, 8, 24) emerges a construction of the extended binary Golay code ${\displaystyle \mathrm {B} _{24))$ that has ${\displaystyle \mathrm {M} _{24))$ as its automorphism group. ${\displaystyle \mathrm {W} _{24))$ generates octads from code words of Hamming weight 8 from the extended binary Golay code, one of five different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24. 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 the subject of the second generation of seven sporadic groups that are subquotients of the automorphism of the Leech lattice, Conway group ${\displaystyle \mathrm {Co} _{0))$.

There are five non-supersingular primes: 37, 43, 53, 61, and 67, all smaller than the largest of fifteen supersingular prime divisors of the friendly giant, 71.[45]

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 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 3125 7776 16807 32768 59049 100000 161051 248832 371293 537824 759375

In decimal

5 is the only prime number to end in the digit 5 in decimal because all other numbers written with a 5 in the ones place are multiples of five, which makes it a 1-automorphic number. 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 ${\displaystyle n}$ raised to the fifth power always ends in the same digit as ${\displaystyle n}$.

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 for the digits 1 to 4. The Kushana and Gupta empires in what is now India had among themselves several different 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 different ways, producing from that were more similar to the digits 4 or 3 than to 5.[46] 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, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa.

Religion and culture

Hinduism

• The god Shiva has five faces[56] and his mantra is also called panchakshari (five-worded) mantra.
• The goddess Saraswati, goddess of knowledge and intellectual is associated with panchami or the number 5.
• 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 most sacred tree in Hinduism has 5 leaves in every leaf stunt.[clarification needed]
• Most of the flowers have 5 petals in them.
• The epic Mahabharata revolves around the battle between Duryodhana and his 99 other brothers and the 5 pandava princes—Dharma, Arjuna, Bhima, Nakula and Sahadeva.

Sikhism

• The five sacred Sikh symbols prescribed by Guru Gobind Singh are commonly known as panj kakars or the "Five Ks" because they start with letter K representing kakka (ਕ) in the Punjabi language's Gurmukhi script. They are: kesh (unshorn hair), kangha (the comb), kara (the steel bracelet), kachhehra (the soldier's shorts), and kirpan (the sword) (in Gurmukhi: ਕੇਸ, ਕੰਘਾ, ਕੜਾ, ਕਛਹਰਾ, ਕਿਰਪਾਨ).[64] Also, there are five deadly evils: kam (lust), krodh (anger), moh (attachment), lobh (greed), and ankhar (ego).

Stations
Series

Sports

• 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).[106]
• In AFL Women's, the top level of women's Australian rules football, each team is allowed 5 "interchanges" (substitute players), who can be freely substituted at any time.
• In baseball scorekeeping, the number 5 represents the third baseman's position.
• The number 5 is used to represent the position of center.
• Each team has five players on the court at a given time. Thus, the phrase "five on five" is commonly used to describe standard competitive basketball.[107]
• The "5-second rule" refers to several related rules designed to promote continuous play. In all cases, violation of the rule results in a turnover.
• Under the FIBA (used for all international play, and most non-US leagues) and NCAA women's rule sets, a team begins shooting bonus free throws once its opponent has committed five personal fouls in a quarter.
• Under the FIBA rules, A player fouls out and must leave the game after committing five fouls
• Five-a-side football is a variation of association football in which each team fields five players.[108]
• In ice hockey:
• A major penalty lasts five minutes.[109]
• There are five different ways that a player can score a goal (teams at even strength, team on the power play, team playing shorthanded, penalty shot, and empty net).[110]
• The area between the goaltender's legs is known as the five-hole.[111]
• In most rugby league competitions, the starting left wing wears this number. An exception is the Super League, which uses static squad numbering.
• In rugby union:

Technology

• 5 is the most common number of gears for automobiles with manual transmission.[113]
• In radio communication, the term "Five by five" is used to indicate perfect signal strength and clarity.[114]
• On almost all devices with a numeric keypad such as telephones, computers, etc., the 5 key has a raised dot or raised bar to make dialing easier. Persons who are blind or have low vision find it useful to be able to feel the keys of a telephone. All other numbers can be found with their relative position around the 5 button (on computer keyboards, the 5 key of the numpad has the raised dot or bar, but the 5 key that shifts with % does not).[115]
• On most telephones, the 5 key is associated with the letters J, K, and L,[116] but on some of the BlackBerry phones, it is the key for G and H.
• The Pentium, coined by Intel Corporation, is a fifth-generation x86 architecture microprocessor.[117]
• The resin identification code used in recycling to identify polypropylene.[118]

Miscellaneous fields

The fives of all four suits in playing cards

Five can refer to:

References

1. Weisstein, Eric W. "5". mathworld.wolfram.com. Retrieved 2020-07-30.
2. ^ Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
3. ^ Weisstein, Eric W. "Mersenne Prime". mathworld.wolfram.com. Retrieved 2020-07-30.
4. ^ Weisstein, Eric W. "Catalan Number". mathworld.wolfram.com. Retrieved 2020-07-30.
5. ^ Weisstein, Eric W. "Twin Primes". mathworld.wolfram.com. Retrieved 2020-07-30.
6. ^ Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
7. ^ Weisstein, Eric W. "Sierpiński Number of the First Kind". mathworld.wolfram.com. Retrieved 2020-07-30.
8. ^ Sloane, N. J. A. (ed.). "Sequence A019434 (Fermat primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21.
9. ^
10. ^ Weisstein, Eric W. "Perrin Sequence". mathworld.wolfram.com. Retrieved 2020-07-30.
11. ^ Tao, Terence (March 2014). "Every odd number greater than 1 is the sum of at most five primes" (PDF). Mathematics of Computation. 83 (286): 997–1038. arXiv:1201.6656. doi:10.1090/S0025-5718-2013-02733-0. S2CID 2618958.
12. ^ Helfgott, Harald Andres (January 2015). "The ternary Goldbach problem". arXiv:1501.05438 [math.NT].
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.
14. ^ 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.
15. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
16. ^ Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 84–86. ISBN 0-387-20860-7.
17. ^ Pomerance, Carl. "On Untouchable Numbers and Related Problems" (PDF). Dartmouth College.
18. ^ Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
19. ^ Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
20. ^ Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-08.
21. ^ 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.
22. ^ 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.
23. ^ William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 2022-07-14.
24. ^ Trigg, C. W. (February 1964). "A Unique Magic Hexagon". Recreational Mathematics Magazine. Retrieved 2022-07-14.
25. ^ Burnstein, Michael (1978). "Kuratowski-Pontrjagin theorem on planar graphs". Journal of Combinatorial Theory, Series B. 24 (2): 228–232. doi:10.1016/0095-8956(78)90024-2.
26. ^ Holton, D. A.; Sheehan, J. (1993). The Petersen Graph. Cambridge University Press. pp. 9.2, 9.5 and 9.9. ISBN 0-521-43594-3.
27. ^ 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. MR 1945373. 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.
28. ^ Royle, G. "Cubic Symmetric Graphs (The Foster Census)." Archived 2008-07-20 at the Wayback Machine
29. ^ de Grey, Aubrey D.N.J. (2018). "The Chromatic Number of the Plane Is at least 5". Geombinatorics. 28: 5–18. arXiv:1804.02385. Bibcode:2016arXiv160407134W.
30. ^ Grünbaum, Branko; Shephard, Geoffrey C. (1987). "Tilings by polygons". Tilings and Patterns. New York: W. H. Freeman and Company. ISBN 978-0-7167-1193-3. MR 0857454. Section 9.3: "Other Monohedral tilings by convex polygons".
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