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.

← 4 5 6 →
−1 0 1 2 3 4 5 6 7 8 9
Numeral systemquinary
Greek numeralΕ´
Roman numeralV, v
Greek prefixpenta-/pent-
Latin prefixquinque-/quinqu-/quint-
Greekε (or Ε)
Arabic, Kurdish٥
Persian, Sindhi, Urdu۵
Chinese numeral

Evolution of the Arabic digitEdit

The evolution of the modern Western digit for the numeral 5 cannot be traced back to the Indian system, as for the digits 1 to 4. The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. The Nagari and Punjabi took these digits and all came up with forms that were similar to a lowercase "h" rotated 180°. The Ghubar Arabs transformed the digit in several ways, producing from that were more similar to the digits 4 or 3 than to 5.[1] It was from those digits that Europeans finally came up with the modern 5.

While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in  .

On the seven-segment display of a calculator, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa.


  is the third smallest prime number, and the second super-prime.[2] It is the first safe prime,[3] the first good prime,[4] the first balanced prime,[5] and the first of three known Wilson primes.[6] Five is the second Fermat prime[2] and the third Mersenne prime exponent,[7] as well as the third Catalan number,[8] and the third Sophie Germain prime.[2] 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).[9][10] It also forms the first pair of sexy primes with 11,[11] which is the fifth prime number and first repunit prime in decimal; a base in-which five is also the first non-trivial 1-automorphic number.[12] Five is the third factorial prime,[13] an alternating factorial,[14] and an Eisenstein prime with no imaginary part and real 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.[15]

The first Pythagorean triple, with a hypotenuse of  

Five is the second Fermat prime of the form  , and more generally the second Sierpiński number of the first kind,  +  .[16] There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.[17] 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 polygons with an odd number of sides that can be constructed purely with a compass and straight-edge, which includes the five-sided regular pentagon.[18][19] 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  -sided polygons, and equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem.[20]

The number 5 is the fifth Fibonacci number, being 2 plus 3.[2] 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), ... (OEISA030452 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.[21]

5 is the third Mersenne prime exponent of the form  , which yields  : 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   = 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.

Every odd number greater than   is the sum of at most five prime numbers, and every odd number greater than   is conjectured to be expressible as the sum of three prime numbers.[22][23] Helfgott has provided a proof of the latter, also known as the odd Goldbach conjecture, that is already widely acknowledged by mathematicians as it still undergoes peer-review.

There are a total of five known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors.[24][25] 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.[26][27] 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  −1(   ) with a   of  , by the Euclid–Euler theorem.[28][29][30] 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.[31][32]

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.[33]

In figurate numbers, 5 is a pentagonal number, with the sequence of pentagonal numbers starting: 1, 5, 12, 22, 35, ...[34]

The factorial of five, or  ! =  , is also 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.[39]

5 is the value of the central cell of the first non-trivial normal magic square, also called the Lo Shu square. Its   x   array of squares has a magic constant   of  , where the sums of its rows, columns, and diagonals are all equal to fifteen.[40] 5 is also the value of the central cell the only non-trivial order-3 normal magic hexagon that is made of nineteen cells.[41]

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   is a solvable group for   , and not for   .

In the Collatz problem, 5 requires five steps to reach 1 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).[42][43] 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. In the analogous 3x − 1 problem, 5 requires five steps to return cyclically to 5, in this instance by multiplying terms by three and subtracting 1 if the terms are odd, and also halving if even: {5 ➙ 14 ➙ 7 ➙ 20 ➙ 10 ➙ 5 ➙ ...}.[44] This is also the first number to generate a cycle that is not trivial (i.e. 1 ➙ 2 ➙ 1 ➙ ...).[45]

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 is a monochromatic subobject isomorphic to  .[46] In general, the Fraïssé limit of a class   of finite relational structure is the age of a countable homogeneous relational structure   iff 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.[47]

Euler's identity,  +   =  , contains five essential numbers used widely in mathematics: Archimedes' constant  , Euler's number  , the imaginary number  , unity  , and zero  , which makes this formula a well known example of beauty in mathematics.[citation needed]

In geometryEdit

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,  . 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. It is often found as a facet inside Islamic Girih tiles, of which there are five different rudimentary types.[48] Generally, star polytopes that are regular only exist in dimensions    <  , and can be constructed using five Miller rules for stellating polyhedra or higher-dimensional polytopes.[49]

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.[50] 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.[51][52] The Petersen graph, which is also a distance-regular graph, is one of only 5 known connected vertex-transitive graphs with no Hamiltonian cycles.[53] The automorphism group of the Petersen graph is the symmetric group   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.[54] 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 also contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations: hexagonal, oblique, rectangular, centered rectangular, and square lattices. Uniform tilings of the plane, furthermore, are generated from combinations of only five regular polygons: the triangle, square, hexagon, octagon, and the dodecagon.[55] The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.[56]

There are five Platonic solids in three-dimensional space: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.[57] 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 five:

The pentatope, or 5-cell, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry   of order 120 = 5! and   group structure. Made of five tetrahedra, its Petrie polygon is a regular pentagon and its orthographic 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.[64]

  • 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.[66]

Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora:  ,  ,  ,  , and  , accompanied by a fifth or sixth general group of unique 4-prisms of Platonic and Archimedean solids. All of these uniform 4-polytopes are generated from 25 uniform polyhedra, which include the five Platonic solids, fifteen Archimedean solids counting two enantiomorphic forms, and five prisms. There are also a total of five Coxeter groups that generate non-prismatic Euclidean honeycombs in 4-space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite facets, as with the order-5 5-cell honeycomb and the order-5 120-cell honeycomb, both of which have five cells around each face. Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10. Likewise, analogues of four-dimensional   hexadecachoric or   icositetrachoric symmetry do not exist in dimensions   ; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have   and   symmetry. There are also five regular projective 4-polytopes in the fourth dimension, all of which are hemi-polytopes of the regular 4-polytopes, with the exception of the 5-cell.[69] Only two regular projective polytopes exist in each higher dimensional space.

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 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   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; no other regular paracompact honeycombs exist in higher dimensions.[70] There are also exclusively twelve complex aperiotopes in   complex spaces of dimensions   ⩾  ; alongside complex polytopes in   and higher under simplex, hypercubic and orthoplex groups (with van Oss polytopes).[71] In particular, a Veronese surface in the projective plane   generalizes a linear condition for a point to be contained inside a conic, which requires five points in the same way that two points are needed to determine a line.[72]

There are five exceptional Lie algebras:  ,  ,  ,  , and  . The smallest of these,  , 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.[73]   is the largest of all five exceptional groups, with the other four as subgroups, and 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.[74] 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 semiregular honeycomb, which includes the three-dimensional alternated cubic honeycomb.[75][76] There are specifically 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   multiply transitive permutation groups on   objects, with   {11, 12, 22, 23, 24}.[77] In particular,  , 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   elements.[78] Of precisely five different conjugacy classes of maximal subgroups of  , one is the almost simple symmetric group   (of order 5!), and another is  , also almost simple, that functions as a point stabilizer which has 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   is sharply 4-transitive,   is sharply 5-transitive and   is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups.[79]   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.[80] All Mathieu groups are subgroups of  , which under the Witt design   of Steiner system   emerges a construction of the extended binary Golay code   that has   as its automorphism group.[81]   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.[82] 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  .[83]

There are five non-supersingular prime numbers — 37, 43, 53, 61, and 67 — all smaller than 71, the largest of fifteen supersingular prime divisors of the friendly giant, itself the largest sporadic group.[84] 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.[85][86] On its own,   can be represented using standard generators   that further dictate a condition where  .[87][88] This condition is also held by other generators that belong to the Tits group  ,[89] the only finite simple group that is a non-strict group of Lie type that can also classify as sporadic. 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  ,[90] which holds the friendly giant as its automorphism group.

List of basic calculationsEdit

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 decimalEdit

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 five sensory organ modalities, with touch represented by the hand's tactility


Religion and cultureEdit


  • The god Shiva has five faces[101] 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.






  • 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).[109] Also, there are five deadly evils: kam (lust), krodh (anger), moh (attachment), lobh (greed), and ankhar (ego).


Other religions and culturesEdit

Art, entertainment, and mediaEdit

Fictional entitiesEdit



  • Modern musical notation uses a musical staff made of five horizontal lines.[126]
  • A scale with five notes per octave is called a pentatonic scale.[127]
  • A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.[128]
  • 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.
  • Using the Latin root, five musicians are called a quintet.[129]
  • Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.







  • 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).[151]
  • 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.
  • In basketball:
    • 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.[152]
    • 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.[153]
  • In ice hockey:
    • A major penalty lasts five minutes.[154]
    • 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).[155]
    • The area between the goaltender's legs is known as the five-hole.[156]
  • 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:


  • 5 is the most common number of gears for automobiles with manual transmission.[158]
  • In radio communication, the term "Five by five" is used to indicate perfect signal strength and clarity.[159]
  • 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).[160]
  • On most telephones, the 5 key is associated with the letters J, K, and L,[161] 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.[162]
  • The resin identification code used in recycling to identify polypropylene.[163]

Miscellaneous fieldsEdit

The fives of all four suits in playing cards

Five can refer to:

See alsoEdit


  1. ^ 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
  2. ^ a b c d e Weisstein, Eric W. "5". mathworld.wolfram.com. Retrieved 2020-07-30.
  3. ^ 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.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ 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.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  7. ^ Weisstein, Eric W. "Mersenne Prime". mathworld.wolfram.com. Retrieved 2020-07-30.
  8. ^ Weisstein, Eric W. "Catalan Number". mathworld.wolfram.com. Retrieved 2020-07-30.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
  11. ^ 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.
  12. ^ 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.
  13. ^ 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.
  14. ^ Weisstein, Eric W. "Twin Primes". mathworld.wolfram.com. Retrieved 2020-07-30.
  15. ^ Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  16. ^ Weisstein, Eric W. "Sierpiński Number of the First Kind". mathworld.wolfram.com. Retrieved 2020-07-30.
  17. ^ Sloane, N. J. A. (ed.). "Sequence A019434 (Fermat primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21.
  18. ^ Conway, John H.; Guy, Richard K. "The Primacy of Primes". The Book of Numbers. New York, NY: Copernicus (Springer). pp. 137–142. doi:10.1007/978-1-4612-4072-3. ISBN 978-1-4612-8488-8. OCLC 32854557. S2CID 115239655.
  19. ^ 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.
  20. ^ 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.
  21. ^ Weisstein, Eric W. "Perrin Sequence". mathworld.wolfram.com. Retrieved 2020-07-30.
  22. ^ 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.
  23. ^ Helfgott, Harald Andres (January 2015). "The ternary Goldbach problem". arXiv:1501.05438 [math.NT].
  24. ^ Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 84–86. ISBN 0-387-20860-7.
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  26. ^ 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.
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