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'''17''' ('''seventeen''') is the [[natural number]] following [[16 (number)|16]] and preceding [[18 (number)|18]]. It is a [[prime number]].
'''17''' ('''seventeen''') is the [[natural number]] following [[16 (number)|16]] and preceding [[18 (number)|18]]. It is a [[prime number]].


17 was described at [[MIT]] as "the least random number", according to the [[Jargon File]].<ref>{{cite web|url=http://www.catb.org/~esr/jargon/html/R/random-numbers.html|title=random numbers|website=catb.org/}}</ref> This is supposedly because, in a study where respondents were asked to choose a random number from 1 to 20, 17 was the most common choice. This study has been repeated a number of times.<ref>{{cite web|url=http://blogs.discovermagazine.com/cosmicvariance/2007/01/30/the-power-of-17/|title=The Power of 17|work=Cosmic Variance|access-date=2010-06-14|archive-date=2008-12-04|archive-url=https://web.archive.org/web/20081204111153/http://blogs.discovermagazine.com/cosmicvariance/2007/01/30/the-power-of-17/|url-status=dead}}</ref>
Seventeen is the sum of the first four prime numbers.


==In mathematics==
== Mathematics ==
17 is a [[Leyland number]]<ref>{{Cite OEIS|A094133|Leyland numbers}}</ref> and [[Leyland number#Leyland primes|Leyland prime]],<ref>{{Cite OEIS|A094133|Leyland prime numbers}}</ref> using 2 & 3 (2<sup>3</sup> + 3<sup>2</sup>). 17 is a [[Leyland number#Leyland number of the second kind|Leyland number of the second kind]]<ref>{{Cite OEIS|A045575|Leyland numbers of the second kind}}</ref> and [[Leyland number#Leyland number of the second kind|Leyland prime of the second kind]],<ref>{{Cite OEIS|A123206|Leyland prime numbers of the second kind}}</ref> using 3 & 4 (3<sup>4</sup> - 4<sup>3</sup>). 17 is a [[Fermat prime]]. 17 is one of six [[lucky numbers of Euler]].<ref>{{Cite OEIS|A014556|Euler's "Lucky" numbers|access-date=2022-11-25}}</ref>
Seventeen is the seventh [[prime number]], which makes it the fourth [[super-prime]],<ref>{{Cite OEIS |A006450 |Prime-indexed primes: primes with prime subscripts. |access-date=2023-06-29 }}</ref> as [[7|seven]] is itself prime.

=== Prime properties ===
Seventeen is the only prime number which is the sum of ''four'' consecutive primes ([[2]], [[3]], [[5]], and [[7]]), as any other four consecutive primes that are added always generate an even number divisible by two.

It forms a [[twin prime]] with [[19 (number)|19]],<ref>{{Cite OEIS |A001359 |Lesser of twin primes |access-date=2022-11-25 }}</ref> a [[cousin prime]] with [[13 (number)|13]],<ref>{{Cite OEIS |A046132 |Larger member p+4 of cousin primes |access-date=2022-11-25 }}</ref> and a [[sexy prime]] with both [[11 (number)|11]] and [[23 (number)|23]].<ref>{{Cite OEIS |A023201 |Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes) |access-date=2022-11-25 }}</ref> Furthermore,

* It is the sixth [[Mersenne prime]] exponent for numbers of the form <math>2^{n} - 1</math>, yielding 131071.<ref>{{Cite OEIS |A000043 |Mersenne exponents |access-date=2022-11-25 }}</ref>
* It is also one of six [[lucky numbers of Euler]] <math>n</math> which produce primes of the form <math>m^{2}-m+n</math> for <math>m=0, \ldots, n-1.</math> (I.e. for <math>n</math> of 17 and <math>m</math> of 16 there is [[257 (number)|257]].)<ref>{{Cite OEIS |A014556 |Euler's "Lucky" numbers |access-date=2022-11-25 }}</ref>
* 17 can be written in the form <math>x^y + y^x</math> and <math>x^y - y^x</math>; and as such, it is a [[Leyland number#Leyland primes|Leyland prime]] (of the first and [[Leyland number#Leyland number of the second kind|second kind]]):<ref>{{Cite OEIS |A094133 |Leyland primes |access-date=2022-11-25 }}</ref><ref>{{Cite OEIS |A045575 |Leyland primes of the second kind |access-date=2022-11-25 }}</ref>
:<math>2^{3} + 3^{2} = 17 = 3^{4} - 4^{3}.</math>

The number of [[integer partition]]s of 17 into prime parts is 17 (the only number <math>n</math> such that its number of such partitions is <math>n</math>).<ref>{{Cite OEIS |A000607 |Number of partitions of n into prime parts. |access-date=2024-02-12 }}</ref>

==== Fermat prime ====
Seventeen is the third [[Fermat prime]], as it is of the form <math>2^{2^{n}} + 1</math> with <math>n = 2</math>.<ref>{{Cite web|url=https://oeis.org/A019434|title=Sloane's A019434 : Fermat primes|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-06-01}}</ref> On the other hand, the seventeenth [[Jacobsthal number|Jacobsthal–Lucas number]] — that is part of a [[sequence]] which includes four Fermat primes (except for [[3]]) — is the fifth and largest known Fermat prime: [[65,537]].<ref>{{Cite OEIS |A014551 |Jacobsthal-Lucas numbers. |access-date=2023-06-29 }}</ref> It is one more than the smallest number with exactly seventeen [[divisor]]s, [[65,536 (number)|65,536]] = 2<sup>16</sup>.<ref>{{Cite OEIS |A005179 |Smallest number with exactly n divisors. |access-date=2023-06-28 }}</ref>


Since seventeen is a Fermat prime, regular [[heptadecagon]]s can be [[constructible polygon|constructed]] with a [[compass]] and unmarked ruler. This was proven by [[Carl Friedrich Gauss]] and ultimately led him to choose mathematics over philology for his studies.<ref>John H. Conway and Richard K. Guy, ''The Book of Numbers''. New York: Copernicus (1996): 11. "Carl Friedrich Gauss (1777–1855) showed that two regular "heptadecagons" (17-sided polygons) could be constructed with ruler and compasses."</ref><ref>[[Theoni Pappas|Pappas, Theoni]], ''Mathematical Snippets'', 2008, p. 42.</ref>
Since seventeen is a Fermat prime, regular [[heptadecagon]]s can be [[constructible polygon|constructed]] with a [[compass]] and unmarked ruler. This was proven by [[Carl Friedrich Gauss]] and ultimately led him to choose mathematics over philology for his studies.<ref>John H. Conway and Richard K. Guy, ''The Book of Numbers''. New York: Copernicus (1996): 11. "Carl Friedrich Gauss (1777–1855) showed that two regular "heptadecagons" (17-sided polygons) could be constructed with ruler and compasses."</ref><ref>[[Theoni Pappas|Pappas, Theoni]], ''Mathematical Snippets'', 2008, p. 42.</ref>


The minimum possible number of givens for a [[sudoku]] puzzle with a unique solution is 17.<ref>{{cite arXiv |eprint=1201.0749 |class=cs.DS |first=Gary |last=McGuire |title=There is no 16-clue sudoku: solving the sudoku minimum number of clues problem |year=2012}}</ref><ref>{{Cite journal |last1=McGuire |first1=Gary |last2=Tugemann |first2=Bastian |last3=Civario |first3=Gilles |date=2014 |title=There is no 16-clue sudoku: Solving the sudoku minimum number of clues problem via hitting set enumeration |journal=Experimental Mathematics |volume=23 |issue=2 |pages=190–217 |doi=10.1080/10586458.2013.870056 |s2cid=8973439}}</ref>
==== Quadratic integer matrix ====
A positive [[Definite quadratic form|definite quadratic]] [[integer matrix]] represents all [[prime number|primes]] when it contains at least the set of seventeen numbers:
:<math>\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73\}.</math>
Only four prime numbers less than the largest member are not part of the set (53, [[59 (number)|59]], 61, and 71).<ref>{{Cite OEIS |A154363 |Numbers from Bhargava's prime-universality criterion theorem }}</ref>


=== Geometric properties ===
=== Geometric properties ===
==== Two-dimensions ====
==== Two-dimensions ====
[[File:Wheel of Theodorus.png|upright=1.45|right|thumb|The [[Spiral of Theodorus]], with a maximum sixteen [[right triangle]]s laid edge-to-edge before one revolution is completed. The largest triangle has a [[hypotenuse]] of <math>\sqrt {17}.</math>]]


*There are seventeen [[Space group#Classification systems|crystallographic space groups]] in two dimensions.<ref>{{Cite OEIS |A006227 |Number of n-dimensional space groups (including enantiomorphs) |access-date=2022-11-25 }}</ref> These are sometimes called [[wallpaper group]]s, as they represent the seventeen possible symmetry types that can be used for [[wallpaper]].
*There are seventeen [[Space group#Classification systems|crystallographic space groups]] in two dimensions.<ref>{{Cite OEIS |A006227 |Number of n-dimensional space groups (including enantiomorphs) |access-date=2022-11-25 }}</ref> These are sometimes called [[wallpaper group]]s, as they represent the seventeen possible symmetry types that can be used for [[wallpaper]].
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*Also in two dimensions, seventeen is the number of combinations of regular polygons that completely [[Vertex (geometry)#Of a plane tiling|fill a plane vertex]].<ref>{{citation|title=The Elements of Plane Practical Geometry, Etc|first=Elmslie William|last=Dallas|publisher=John W. Parker & Son|year=1855|page=134|url=https://books.google.com/books?id=y4BaAAAAcAAJ&pg=PA134}}.</ref> Eleven of these belong to [[Euclidean tilings of convex regular polygons#Regular tilings|regular and semiregular tilings]], while 6 of these (3.7.42,<ref>{{Cite web|url=http://gruze.org/tilings/3_7_42_shield|title=Shield - a 3.7.42 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> [[Icositetragon#Related polygons|3.8.24]],<ref>{{Cite web|url=http://gruze.org/tilings/dancer|title=Dancer - a 3.8.24 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> [[Octadecagon#Uses|3.9.18]],<ref>{{Cite web|url=http://gruze.org/tilings/3_9_18_art|title=Art - a 3.9.18 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> [[Pentadecagon#Uses|3.10.15]],<ref>{{Cite web|url=http://gruze.org/tilings/3_10_15_fighters|title=Fighters - a 3.10.15 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> [[Icosagon#Uses|4.5.20]],<ref>{{Cite web|url=http://gruze.org/tilings/compass|title=Compass - a 4.5.20 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> and 5.5.10)<ref>{{Cite web|url=http://gruze.org/tilings/5_5_10_broken_roses|title=Broken roses - three 5.5.10 tilings|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> exclusively surround a point in the plane and fill it only when irregular polygons are included.<ref>{{Cite web|url=https://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/|title=Pentagon-Decagon Packing|website=American Mathematical Society|publisher=AMS|access-date=2022-03-07}}</ref>
*Also in two dimensions, seventeen is the number of combinations of regular polygons that completely [[Vertex (geometry)#Of a plane tiling|fill a plane vertex]].<ref>{{citation|title=The Elements of Plane Practical Geometry, Etc|first=Elmslie William|last=Dallas|publisher=John W. Parker & Son|year=1855|page=134|url=https://books.google.com/books?id=y4BaAAAAcAAJ&pg=PA134}}.</ref> Eleven of these belong to [[Euclidean tilings of convex regular polygons#Regular tilings|regular and semiregular tilings]], while 6 of these (3.7.42,<ref>{{Cite web|url=http://gruze.org/tilings/3_7_42_shield|title=Shield - a 3.7.42 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> [[Icositetragon#Related polygons|3.8.24]],<ref>{{Cite web|url=http://gruze.org/tilings/dancer|title=Dancer - a 3.8.24 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> [[Octadecagon#Uses|3.9.18]],<ref>{{Cite web|url=http://gruze.org/tilings/3_9_18_art|title=Art - a 3.9.18 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> [[Pentadecagon#Uses|3.10.15]],<ref>{{Cite web|url=http://gruze.org/tilings/3_10_15_fighters|title=Fighters - a 3.10.15 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> [[Icosagon#Uses|4.5.20]],<ref>{{Cite web|url=http://gruze.org/tilings/compass|title=Compass - a 4.5.20 tiling|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> and 5.5.10)<ref>{{Cite web|url=http://gruze.org/tilings/5_5_10_broken_roses|title=Broken roses - three 5.5.10 tilings|website=Kevin Jardine's projects|publisher=Kevin Jardine|access-date=2022-03-07}}</ref> exclusively surround a point in the plane and fill it only when irregular polygons are included.<ref>{{Cite web|url=https://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/|title=Pentagon-Decagon Packing|website=American Mathematical Society|publisher=AMS|access-date=2022-03-07}}</ref>


*Seventeen is the minimum number of [[Vertex (geometry)|vertices]] on a two-dimensional [[Graph (discrete mathematics)|graph]] such that, if the [[Edge (geometry)|edges]] are colored with three different colors, there is bound to be a [[monochromatic triangle]]; see [[Ramsey's theorem#A multicolour example: R(3, 3, 3) = 17|Ramsey's theorem]].<ref>{{Cite OEIS |A003323 |Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's. |access-date=2022-11-25 }}</ref>
*On the other hand, either 16 or 18 [[unit square]]s can be formed into rectangles with perimeter equal to the area; and there are no other [[natural number]]s with this property. The [[Platonist]]s regarded this as a sign of their peculiar propriety; and [[Plutarch]] notes it when writing that the [[Pythagoreans]] "utterly abominate" 17, which "bars them off from each other and disjoins them".<ref>{{Cite book|last=Babbitt|first=Frank Cole|title=Plutarch's Moralia|publisher=Loeb|year=1936|volume=V|url=https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/Isis_and_Osiris*/C.html#42}}</ref>


*Either 16 or 18 [[unit square]]s can be formed into rectangles with perimeter equal to the area; and there are no other [[natural number]]s with this property. The [[Platonist]]s regarded this as a sign of their peculiar propriety; and [[Plutarch]] notes it when writing that the [[Pythagoreans]] "utterly abominate" 17, which "bars them off from each other and disjoins them".<ref>{{Cite book|last=Babbitt|first=Frank Cole|title=Plutarch's Moralia|publisher=Loeb|year=1936|volume=V|url=https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/Isis_and_Osiris*/C.html#42}}</ref>
Seventeen is the minimum number of [[Vertex (geometry)|vertices]] on a two-dimensional [[Graph (discrete mathematics)|graph]] such that, if the [[Edge (geometry)|edges]] are colored with three different colors, there is bound to be a [[monochromatic triangle]]; see [[Ramsey's theorem#A multicolour example: R.283.2C3.2C3.29 .3D 17|Ramsey's theorem]].<ref>{{Cite OEIS |A003323 |Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's. |access-date=2022-11-25 }}</ref>

17 is the least <math>k</math> for the [[Spiral of Theodorus|Theodorus Spiral]] to complete one [[revolution]].<ref>{{Cite OEIS |A072895 |Least k for the Theodorus spiral to complete n revolutions |access-date=2024-06-19 }}</ref> This, in the sense of [[Plato]], who questioned why Theodorus (his tutor) stopped at <math>\sqrt{17}</math> when illustrating adjacent [[right triangle]]s whose bases are [[1|unit]]s and heights are successive [[square root]]s, starting with <math>1</math>. In part due to Theodorus’s work as outlined in Plato’s ''[[Theaetetus (dialogue)|Theaetetus]]'', it is believed that Theodorus had proved all the square roots of non-[[Square number|square]] integers from [[3]] to 17 are [[Irrational number|irrational]] by means of this spiral.


==== Enumeration of icosahedron stellations ====
==== Enumeration of icosahedron stellations ====
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Seventeen is the highest dimension for [[Coxeter-Dynkin diagram#Hypercompact Coxeter groups (Vinberg polytopes)|paracompact Vineberg polytopes]] with rank <math>n+2</math> mirror [[Facet (geometry)|facets]], with the lowest belonging to the third.<ref>{{cite journal |last=Tumarkin |first=P.V. |date=May 2004 |title=Hyperbolic Coxeter N-Polytopes with n+2 Facets |journal=Mathematical Notes |url=https://doi.org/10.1023/B:MATN.0000030993.74338.dd |volume=75 |issue=5/6 |pages=848–854 |doi=10.1023/B:MATN.0000030993.74338.dd |arxiv=math/0301133 |access-date=18 March 2022}}</ref>
Seventeen is the highest dimension for [[Coxeter-Dynkin diagram#Hypercompact Coxeter groups (Vinberg polytopes)|paracompact Vineberg polytopes]] with rank <math>n+2</math> mirror [[Facet (geometry)|facets]], with the lowest belonging to the third.<ref>{{cite journal |last=Tumarkin |first=P.V. |date=May 2004 |title=Hyperbolic Coxeter N-Polytopes with n+2 Facets |journal=Mathematical Notes |url=https://doi.org/10.1023/B:MATN.0000030993.74338.dd |volume=75 |issue=5/6 |pages=848–854 |doi=10.1023/B:MATN.0000030993.74338.dd |arxiv=math/0301133 |access-date=18 March 2022}}</ref>


17 is the seventh ''[[Supersingular prime (moonshine theory)|supersingular prime]]'' that divides the [[Order (group theory)|order]] of six [[sporadic group]]s ([[Janko group J3|''J<sub>3</sub>'']], [[Held group|''He'']], [[Fischer group Fi23|''Fi<sub>23</sub>'']], [[Fischer group Fi24|''Fi<sub>24</sub>'']], [[Baby monster group|''B'']], and [[Monster group|''F<sub>1</sub>'']]) inside the [[Sporadic group#Happy Family|Happy Family]] of such groups.<ref>{{Cite OEIS |A002267 |The 15 supersingular primes |access-date=2022-11-25 }}</ref> The 16th and 18th prime numbers ([[53 (number)|53]] and [[61 (number)|61]]) are the only two primes less than [[71 (number)|71]] that do not divide the [[Order (group theory)|order]] of any sporadic group including the [[Pariah group|pariahs]], with this prime as the largest such supersingular prime that divides the largest of these groups (''F<sub>1</sub>''). On the other hand, if the [[Tits group]] is included as a ''non-strict'' group of [[Group of Lie type|Lie type]], then there are seventeen total classes of [[Lie group]]s that are simultaneously [[Finite group|finite]] and [[Simple group|simple]] (see, [[classification of finite simple groups]]). In [[base ten]], (17, 71) form the seventh permutation class of [[permutable prime]]s.<ref>{{Cite OEIS |A258706 |Absolute primes: every permutation of digits is a prime. Only the smallest representative of each permutation class is shown. |access-date=2023-06-29 }}</ref>
17 is a [[Supersingular prime (moonshine theory)|supersingular prime]], because it divides the order of the [[Monster group]].<ref>{{Cite OEIS |A002267 |The 15 supersingular primes |access-date=2022-11-25 }}</ref> If the [[Tits group]] is included as a ''non-strict'' group of [[Group of Lie type|Lie type]], then there are seventeen total classes of [[Lie group]]s that are simultaneously [[Finite group|finite]] and [[Simple group|simple]] (see [[classification of finite simple groups]]). In [[base ten]], (17, 71) form the seventh permutation class of [[permutable prime]]s.<ref>{{Cite OEIS |A258706 |Absolute primes: every permutation of digits is a prime. Only the smallest representative of each permutation class is shown. |access-date=2023-06-29 }}</ref>


=== Other notable properties ===
=== Other notable properties ===
* The sequence of residues (mod {{mvar|n}}) of a [[Googol#Properties|googol]] and [[Googolplex#Mod n|googolplex]], for <math>n=1, 2, 3, ...</math>, agree up until <math>n=17</math>.
* The sequence of residues (mod {{mvar|n}}) of a [[Googol#Properties|googol]] and [[Googolplex#Mod n|googolplex]], for <math>n=1, 2, 3, ...</math>, agree up until <math>n=17</math>.{{cn|date=August 2024}}
* Seventeen is the longest sequence for which a solution exists in the [[irregularity of distributions]] problem.<ref>{{cite journal|author1=[[Elwyn Berlekamp|Berlekamp, E. R.]] |author2=[[Ronald L. Graham|Graham, R. L.]] |title=Irregularities in the distributions of finite sequences | journal = [[Journal of Number Theory]]|volume=2|year=1970|issue=2 |pages=152–161|mr=0269605|doi=10.1016/0022-314X(70)90015-6|bibcode=1970JNT.....2..152B |doi-access=free}}</ref>
* Seventeen is the longest sequence for which a solution exists in the [[irregularity of distributions]] problem.<ref>{{cite journal|author1=[[Elwyn Berlekamp|Berlekamp, E. R.]] |author2=[[Ronald L. Graham|Graham, R. L.]] |title=Irregularities in the distributions of finite sequences | journal = [[Journal of Number Theory]]|volume=2|year=1970|issue=2 |pages=152–161|mr=0269605|doi=10.1016/0022-314X(70)90015-6|bibcode=1970JNT.....2..152B |doi-access=free}}</ref>


==== Complex analysis ====
== In science==
[[File:Standard Model of Elementary Particles.svg|right|240px|thumb|The [[elementary particle]]s in the [[Standard Model]] of physics ]]
There are seventeen orthogonal curvilinear [[coordinate systems]] (to within a conformal symmetry) in which the three-variable [[Laplace equation]] can be solved using the [[separation of variables]] technique.

==== Sudoku puzzle ====
The minimum possible number of givens for a [[sudoku]] puzzle with a unique solution is 17.<ref>{{cite arXiv|last=McGuire|first=Gary|title=There is no 16-clue sudoku: solving the sudoku minimum number of clues problem|eprint=1201.0749|class=cs.DS|year=2012}}</ref><ref>{{Cite journal |last1=McGuire |first1=Gary |last2=Tugemann |first2=Bastian |last3=Civario |first3=Gilles |s2cid=8973439 |date=2014 |title=There is no 16-clue sudoku: Solving the sudoku minimum number of clues problem via hitting set enumeration |journal=Experimental Mathematics |volume=23 |issue=2 |pages=190–217 |doi=10.1080/10586458.2013.870056 }}</ref>

=== Spiral of Theodorus ===
[[File:Wheel of Theodorus.png|300px|right|thumb|The [[Spiral of Theodorus]], with a maximum sixteen [[right triangle]]s laid edge-to-edge before one revolution is completed. The largest triangle has a [[hypotenuse]] of <math>\sqrt {17}</math>.]]

17 is the least <math>k</math> for the [[Spiral of Theodorus|Theodorus Spiral]] to complete one [[revolution]].<ref>{{Cite OEIS |A072895 |Least k for the Theodorus spiral to complete n revolutions |access-date=2024-06-19 }}</ref> This, in the sense of [[Plato]], who questioned why Theodorus (his tutor) stopped at <math>\sqrt{17}</math> when illustrating adjacent [[right triangle]]s whose bases are [[1|unit]]s and heights are successive [[square root]]s, starting with <math>1</math>. In part due to Theodorus’s work as outlined in Plato’s ''[[Theaetetus (dialogue)|Theaetetus]]'', it is believed that Theodorus had proved all the square roots of non-[[Square number|square]] integers from [[3]] to 17 are [[Irrational number|irrational]] by means of this spiral.

==In science==


=== Physics ===
=== Physics ===
Seventeen is the number of [[elementary particle]]s with unique names in the [[Standard Model]] of physics.<ref>{{cite journal|url=http://physics.info/standard/|title=The Standard Model|author=Glenn Elert|journal=The Physics Hypertextbook|year=2021}}</ref>
Seventeen is the number of [[elementary particle]]s with unique names in the [[Standard Model]] of physics.<ref>{{cite journal|url=http://physics.info/standard/|title=The Standard Model|author=Glenn Elert|journal=The Physics Hypertextbook|year=2021}}</ref>


It is more specifically the number of [[Particle physics|particles]] that are either their own [[antiparticle]] (i.e. the five classes of [[Boson#Elementary bosons|scalar or vector bosons]]) and those that are not (collectively, twelve [[quark]]s and [[lepton]]s); that is, without distinguishing between eight [[gluon]]s, or two [[W boson|W<sup>±</sup> bosons]] (that are each other's antiparticles, W<sup>+</sup> and W<sup>−</sup>).
=== Chemistry ===
=== Chemistry ===
[[Group (periodic table)|Group 17]] of the [[periodic table]] is called the [[halogens]]. The [[atomic number]] of [[chlorine]] is 17.
[[Group (periodic table)|Group 17]] of the [[periodic table]] is called the [[halogens]]. The [[atomic number]] of [[chlorine]] is 17.


==In languages==
=== Biology ===
Some [[Periodical cicadas|species]] of [[cicada]]s have a life cycle of 17 years (i.e. they are buried in the ground for 17 years between every mating season).


===Grammar===
== In religion ==
* In the [[Yasna]] of [[Zoroastrianism]], seventeen chapters were written by [[Zoroaster]] himself. These are the five [[Gathas]].
In Catalan, 17 is the first compound number ({{lang|ca|disset}}). The numbers 11 ({{lang|ca|onze}}) through 16 ({{lang|ca|setze}}) have their own names.
* The number of [[sura]]t [[al-Isra]] in the [[Qur'an]] is seventeen, at times included as one of seven [[Al-Musabbihat]]. 17 is the total number of [[Rak'a#Daily prayers|Rak'a]]s that Muslims perform during [[Salat]] on a daily basis.


== Other fields ==
In French, 17 is the first compound number ({{lang|fr|dix-sept}}). The numbers 11 ({{lang|fr|onze}}) through 16 ({{lang|fr|seize}}) have their own names.
'''Seventeen''' is:
* The total number of syllables in a [[haiku]] (5 + 7 + 5).
* The maximum number of strokes of a [[radical (Chinese character)|Chinese radical]].


==Age 17==
=== Music ===
* In most countries across the world, it is the last age at which one is considered a [[Minor (law)|minor]] under law.
* In the UK, the minimum age for taking [[driver's education|driving lessons]], and to drive a car or a van
* In the US and Canada, it is the age at which one may purchase, rent, or reserve [[Entertainment Software Rating Board#Restricted ratings|M-rated]] video games without parental consent
* In some US states,<ref>{{cite web|url=http://www.age-of-consent.info/|archive-url=https://web.archive.org/web/20110417024317/http://www.age-of-consent.info/|url-status=dead|archive-date=2011-04-17|title=Age Of Consent By State}}</ref> and some jurisdictions around the world, 17 is the [[age of consent|age of sexual consent]]<ref>{{cite web|url=http://www.avert.org/age-of-consent.htm|title=Age of consent for sexual intercourse|date=2015-06-23}}</ref>
* In most US states, Canada and in the UK, the age at which one may [[donate blood]] (without parental consent)
* In many countries and jurisdictions, the age at which one may obtain a [[driver's license]]
* In the US, the age at which one may watch, rent, or purchase [[Motion Picture Association of America film rating system|R-rated]] movies without parental consent
*The U.S. [[TV Parental Guidelines]] system sets 17 as the minimum age one can watch programs with a TV-MA rating without parental guidance.
* In the US, the age at which one can enlist in the armed forces with parental consent
* In the US, the age at which one can apply for a [[private pilot licence]] for powered flight (however, applicants can obtain a student pilot certificate at age 16)
* In Greece and Indonesia, the voting age
* In Chile and Indonesia, the minimum driving age.
* In [[Tajikistan]], [[North Korea]] and [[Timor-Leste]], the [[age of majority]]


Where [[Pythagoreanism|Pythagoreans]] saw 17 in between 16 from its [[Epogdoon]] of 18 in distaste,<ref>{{cite book|url=https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/Isis_and_Osiris*/C.html|author=Plutarch, Moralia|title=Isis and Osiris (Part 3 of 5)|publisher= Loeb Classical Library edition|date=1936}}</ref> the ratio 18:17 was a popular approximation for the [[equal temperament|equal tempered]] [[semitone]] (12-tone) during the [[Renaissance]].
==In culture==
===Music===
{{main|17 (disambiguation)#Music}}


====Bands====
== Notes ==
{{Notelist}}
* [[17 Hippies]], a Worldmusic band, based in Berlin
* [[Seventeen (South Korean band)|Seventeen]] ({{Lang|ko|세븐틴}}), a South Korean boy band
* [[Heaven 17]], an English new wave band
* [[East 17]], an English boy band

====Albums====
* [[17 (XXXTentacion album)|''17'' (XXXTentacion album)]]
* [[17 (Motel album)|''17'' (Motel album)]]
* [[17 (Ricky Martin album)|''17'' (Ricky Martin album)]]
* ''[[Chicago 17]]'', a 1984 album by Chicago
* ''[[Seventeen Days]]'', a 2005 album by 3 Doors Down
* ''[[Seventeen Seconds]]'', a 1980 album by the Cure
* ''17 Carat'', a 2015 EP by [[Seventeen (South Korean band)|Seventeen]]
* ''Sector 17'', a 2022 repackaged album by [[Seventeen (South Korean band)|Seventeen]]

====Songs====
* "17 Again", a song by [[Tide Lines]]
* [[17 (Sky Ferreira song)|"17" (Sky Ferreira song)]]
* [[17 (Yourcodenameis:Milo song)|"17" (Yourcodenameis:Milo song)]]
* "[[17 Again (song)|17 Again]]", a song by Eurythmics
* "[[17 år]]", a song by Veronica Maggio
* "17 Crimes", a song by [[AFI (band)|AFI]]
* "[[17 Days (song)|17 Days]]", a song by Prince
* "17", a song by [[Dan Bălan]]
* "17", a song by [[Jethro Tull (band)|Jethro Tull]]
* "17", a song by [[Kings of Leon]]
* "17", a song by [[Milburn (band)|Milburn]]
* "17", a song by Rick James from ''[[Reflections (Rick James album)|Reflections]]''
* "17", a B-side by [[Shiina Ringo]] on the "Tsumi to Batsu" single
* "17", a song by [[the Smashing Pumpkins]] from the album ''[[Adore (The Smashing Pumpkins album)|Adore]]''
* "17", a song by [[Youth Lagoon]] from the album ''[[The Year of Hibernation]]''
* "17 Days", a song by [[Prince & the Revolution]], B side from the 1984 "When Doves Cry" single
* [[Seventeen (Jet song)|"Seventeen" (Jet song)]]
* [[Seventeen (Ladytron song)|"Seventeen" (Ladytron song)]]
* [[Seventeen (Winger song)|"Seventeen" (Winger song)]]
* "Seventeen", a song by ¡Forward, Russia! from ''[[Give Me a Wall]]''
* "Seventeen", a song by Jimmy Eat World from ''[[Static Prevails]]''
* "Seventeen", a song by Marina & the Diamonds from the US edition of ''[[The Family Jewels (Marina and the Diamonds album)|The Family Jewels]]''
* "Seventeen", a song by Mat Kearney from the iTunes edition of ''[[Young Love (Mat Kearney album)|Young Love]]''
* "Seventeen", a song from the [[Repo! The Genetic Opera (soundtrack)|''Repo! The Genetic Opera'' soundtrack]]
* "Seventeen", the original title of the song "[[I Saw Her Standing There]]" by [[The Beatles]]
* "Seventeen", a song by the [[Sex Pistols]] from ''[[Never Mind the Bollocks, Here's the Sex Pistols]]''
* "[[Seventeen Forever]]", a song by Metro Station
* "[[At Seventeen]]", a song by Janis Ian
* "[[Edge of Seventeen]]", a song by Stevie Nicks
* "Seventeen Ain't So Sweet", a song by The Red Jumpsuit Apparatus from ''[[Don't You Fake It]]''
* "Only 17", a song by [[Rucka Rucka Ali]]
* "Opus 17 (Don't You Worry 'Bout Me)", a song by [[Frankie Valli and the Four Seasons]]
* "(She's) Sexy + 17", a song by Stray Cats from ''[[Rant N' Rave with the Stray Cats]]''
* "Hello, Seventeen", a song by [[12012]]
* "Section 17 (Suitcase Calling)", a song by [[The Polyphonic Spree]]
* "Day Seventeen: Accident?", a song by [[Ayreon]]
* "Seventeen", a song by [[Alessia Cara]]
* "Seventeen", a song performed by [[Marina and the Diamonds]]
* "Seventeen" and "Seventeen (Reprise)", songs in the musical ''[[Heathers: The Musical|Heathers]]''
* "Seventeen" and "Seventeen (Reprise)", songs in the musical ''[[Tuck Everlasting (musical)|Tuck Everlasting]]''

====Other====
* [[Seventeen (musical)|''Seventeen'']], a 1951 American musical
* The ratio 18:17 was a popular approximation for the [[equal temperament|equal tempered]] [[semitone]] during the Renaissance

===Film===
* ''[[Seventeen (1916 film)|Seventeen]]'' (1916), an adaptation of the [[Seventeen (Tarkington novel)|novel of the same name]] by [[Booth Tarkington]]
* ''[[Number 17 (1928 film)|Number 17]]'' (1928), a British-German film
* ''[[Number Seventeen]]'' (1932), directed by [[Alfred Hitchcock]]
* ''[[Seventeen (1940 film)|Seventeen]]'' (1940), a second adaptation of the Tarkington novel
* ''[[Number 17 (1949 film)|Number 17]]'' (1949), a Swedish film
* ''[[Stalag 17]]'' (1953), directed by [[Billy Wilder]]
* ''[[All I Want (film)|Try Seventeen]]'' (2002), directed by Jeffrey Porter
* '' [[17 Again (film)|17 Again]]'' (2009), directed by [[Burr Steers]]

===Anime and manga===
* [[Android 17]], a character from the ''[[Dragon Ball]]'' series
* Detective Konawaka from the [[Paprika (anime)|''Paprika'']] anime has a strong dislike for the number 17

===Games===
* The computer game ''[[Half-Life 2]]'' takes place in and around [[City 17]]
* The visual novel ''[[Ever 17: The Out of Infinity]]'' strongly revolves around the number 17

===Print===
* The title of ''[[Seventeen (American magazine)|Seventeen]]'', a magazine
* The title of ''[[Just Seventeen]]'', a former magazine
* The number 17 is a recurring theme in the works of [[novelist]] [[Steven Brust]]. All of his chaptered novels have either 17 chapters or two books of 17 chapters each. Multiples of 17 frequently appear in his novels set in the fantasy world of [[Dragaera]], where the number is considered holy.
* In ''[[The Illuminatus! Trilogy]]'', the symbol for [[Discordianism]] includes a pyramid with 17 steps because 17 has "virtually no interesting geometric, arithmetic, or mystical qualities". However, for the [[Illuminati]], 17 is tied with the "[[23 (numerology)|23/17 phenomenon]]".
* In the [[Harry Potter universe]]
** 17 is the coming of age for wizards. It is equivalent to the usual coming of age at 18.
** 17 is the number of Sickles in one Galleon in the [[Harry Potter Universe#Coins|British wizards' currency]].

===Religion===
* According to [[Plutarch]]'s [[Moralia]], the Egyptians have a legend that the end of Osiris' life came on the seventeenth of a month, on which day it is quite evident to the eye that the period of the full moon is over. Now, because of this, the Pythagoreans call this day "the Barrier", and utterly abominate this number. For the number seventeen, coming in between the square sixteen and the oblong rectangle eighteen, which, as it happens, are the only plane figures that have their perimeters equal their areas, bars them off from each other and disjoins them, and breaks up the [[epogdoon]] by its division into unequal intervals.<ref>{{cite book|url=https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/Isis_and_Osiris*/C.html|author=Plutarch, Moralia|title=Isis and Osiris (Part 3 of 5)|publisher= Loeb Classical Library edition|date=1936}}</ref>
* In the [[Yasna]] of [[Zoroastrianism]], seventeen chapters were written by [[Zoroaster]] himself. These are the [[Gathas]].
* The number of the [[raka'ah]]s that Muslims perform during [[Salat]] on a daily basis.
* The number of [[sura]]t [[al-Isra]] in the [[Qur'an]].

==In sports==

* 17 is the number of the longest winning streak in NHL history, which the [[Pittsburgh Penguins]] achieved in 1993.
* [[Larry Ellison]]'s victorious 2013 [[Americas Cup]] Oracle racing yacht bears the name "17".
* 17 is the number of individual laws mentioned in the [[Laws of the Game (association football)]].
* 17 is the number of games played by each [[NFL]] team as of 2021.
* Since the start of the [[2014 Formula One World Championship|2014 season]], [[Formula One]] [[List of Formula One drivers|drivers]] have been able to choose [[List of Formula One driver numbers|their own car number]]; however, following the fatal accident of [[Jules Bianchi]], who drove car #17, the number was retired.

==In other fields==
'''Seventeen''' is:
* Described at [[MIT]] as 'the least random number', according to the [[Jargon File]].<ref>{{cite web|url=http://www.catb.org/~esr/jargon/html/R/random-numbers.html|title=random numbers|website=catb.org/}}</ref> This is supposedly because in a study where respondents were asked to choose a random number from 1 to 20, 17 was the most common choice.
** This study has been repeated a number of times.<ref>{{cite web|url=http://blogs.discovermagazine.com/cosmicvariance/2007/01/30/the-power-of-17/|title=The Power of 17|work=Cosmic Variance|access-date=2010-06-14|archive-date=2008-12-04|archive-url=https://web.archive.org/web/20081204111153/http://blogs.discovermagazine.com/cosmicvariance/2007/01/30/the-power-of-17/|url-status=dead}}</ref>
* The number of guns in a 17-gun [[salute]] to U.S. Army, Air Force and Marine Corps Generals, and Navy and Coast Guard admirals.
* The maximum number of strokes of a [[radical (Chinese character)|Chinese radical]].
* The total number of syllables in a [[haiku]] (5 + 7 + 5).
* In the [[Nordic countries]] the seventeenth day of the year is considered the ''heart'' and/or the ''back'' of winter.
* "Highway 17" or "Route 17": See [[List of highways numbered 17]] and [[List of public transport routes numbered 17]].
* Seventeen, also known as Lock Seventeen, an unincorporated place in [[Clay Township, Tuscarawas County, Ohio]].
* ''Seventeen'' was the former name of a yacht prior to being commissioned in the [[US Navy]] as the {{USS|Carnelian|PY-19}}.
[[Image:Alitalia-17.jpg|thumb|No row 17 in [[Alitalia]] planes]]
* In [[Italian culture]], the number 17 is considered unlucky. When viewed as the Roman numeral, XVII, it is then changed anagrammatically to VIXI, which in the [[Latin language]] translates to "I lived", the [[perfect (grammar)|perfect]] implying "My life is over." (c.f. "''Vixerunt''", [[Cicero]]'s famous announcement of an execution.) [[Renault]] sold its "[[Renault 15/17|R17]]" model in Italy as "R177". See [[Cesana Pariol]] in the sport section about the name of curve 17.
* The fear of the number 17 is called '[[heptadecaphobia]]' or 'heptakaidekaphobia'.
* Some [[Periodical cicadas|species]] of [[cicada]]s have a life cycle of 17 years (i.e. they are buried in the ground for 17 years between every mating season).
* The number to call police in France.
* [[Force 17]], a special operations unit of the Palestinian Fatah movement.
* The number of the French department [[Charente-Maritime]].
* [[Malaysia Airlines Flight 17]] was [[List of aircraft shootdowns|shot down]] by Russian-controlled forces on 17 July 2014 after flying over eastern Ukraine. The first test flight of the plane, a [[Boeing 777-200ER]], was on 17 July 1997, exactly 17 years prior to the doomed flight.


==References==
== References ==
<references/>
<references/>
* {{cite journal|author1=Berlekamp, E. R. |author2-link=Ronald L. Graham |author2=Graham, R. L. |title=Irregularities in the distributions of finite sequences
* {{cite journal|author1=Berlekamp, E. R. |author2-link=Ronald L. Graham |author2=Graham, R. L. |title=Irregularities in the distributions of finite sequences
| journal = [[Journal of Number Theory]]|volume=2|year=1970|pages=152–161|mr=0269605|doi=10.1016/0022-314X(70)90015-6|issue=2|bibcode=1970JNT.....2..152B|author1-link=Elwyn Berlekamp |doi-access=free}}
| journal = [[Journal of Number Theory]]|volume=2|year=1970|pages=152–161|mr=0269605|doi=10.1016/0022-314X(70)90015-6|issue=2|bibcode=1970JNT.....2..152B|author1-link=Elwyn Berlekamp |doi-access=free}}


==External links==
== External links ==
{{Commons category}}
{{Commons category}}
{{wiktionary|seventeen}}
{{wiktionary|seventeen}}

* [http://www.vinc17.org/d17_eng.html Properties of 17]
* [https://primes.utm.edu/curios/page.php/17.html Prime Curios!: 17]
* [http://www.yellowpigs.net/index.php?topic=yellowpigs/YP_seventeen Mathematical properties of 17] {{Webarchive|url=https://web.archive.org/web/20190729010229/https://www.yellowpigs.net/index.php?topic=yellowpigs%2FYP_seventeen |date=2019-07-29 }} at yellowpigs.net
* [https://web.archive.org/web/20100705142954/http://www.hilmar-alquiros.de/siebzehn.htm 17]
* [https://web.archive.org/web/20110818122755/http://scienceblogs.com/cognitivedaily/2007/02/is_17_the_most_random_number.php Is 17 the "most random" number?]
* [https://web.archive.org/web/20110818122755/http://scienceblogs.com/cognitivedaily/2007/02/is_17_the_most_random_number.php is 17 the most random number] at the wayback machine.
* [https://www.VirtueScience.com/17.html Number 17 at the Database of Number Correlations]
* [https://primes.utm.edu/curios/page.php/17.html Prime Curios for the number 17]


{{Integers|zero}}
{{Integers|zero}}

Latest revision as of 11:37, 30 October 2024

For other uses, see 17.
← 16 17 18 →
Cardinalseventeen
Ordinal17th
(seventeenth)
Numeral systemseptendecimal
Factorizationprime
Prime7th
Divisors1, 17
Greek numeralΙΖ´
Roman numeralXVII
Binary100012
Ternary1223
Senary256
Octal218
Duodecimal1512
Hexadecimal1116
Hebrew numeralי"ז
Babylonian numeral𒌋𒐛

17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.

17 was described at MIT as "the least random number", according to the Jargon File.[1] This is supposedly because, in a study where respondents were asked to choose a random number from 1 to 20, 17 was the most common choice. This study has been repeated a number of times.[2]

Mathematics

[edit]

17 is a Leyland number[3] and Leyland prime,[4] using 2 & 3 (23 + 32). 17 is a Leyland number of the second kind[5] and Leyland prime of the second kind,[6] using 3 & 4 (34 - 43). 17 is a Fermat prime. 17 is one of six lucky numbers of Euler.[7]

Since seventeen is a Fermat prime, regular heptadecagons can be constructed with a compass and unmarked ruler. This was proven by Carl Friedrich Gauss and ultimately led him to choose mathematics over philology for his studies.[8][9]

The minimum possible number of givens for a sudoku puzzle with a unique solution is 17.[10][11]

Geometric properties

[edit]

Two-dimensions

[edit]
The Spiral of Theodorus, with a maximum sixteen right triangles laid edge-to-edge before one revolution is completed. The largest triangle has a hypotenuse of
  • Either 16 or 18 unit squares can be formed into rectangles with perimeter equal to the area; and there are no other natural numbers with this property. The Platonists regarded this as a sign of their peculiar propriety; and Plutarch notes it when writing that the Pythagoreans "utterly abominate" 17, which "bars them off from each other and disjoins them".[22]

17 is the least for the Theodorus Spiral to complete one revolution.[23] This, in the sense of Plato, who questioned why Theodorus (his tutor) stopped at when illustrating adjacent right triangles whose bases are units and heights are successive square roots, starting with . In part due to Theodorus’s work as outlined in Plato’s Theaetetus, it is believed that Theodorus had proved all the square roots of non-square integers from 3 to 17 are irrational by means of this spiral.

Enumeration of icosahedron stellations

[edit]

In three-dimensional space, there are seventeen distinct fully supported stellations generated by an icosahedron.[24] The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules.[25][26] Without counting the icosahedron as a zeroth stellation, this total becomes 58, a count equal to the sum of the first seven prime numbers (2 + 3 + 5 + 7 ... + 17).[27] Seventeen distinct fully supported stellations are also produced by truncated cube and truncated octahedron.[24]

Four-dimensional zonotopes

[edit]

Seventeen is also the number of four-dimensional parallelotopes that are zonotopes. Another 34, or twice 17, are Minkowski sums of zonotopes with the 24-cell, itself the simplest parallelotope that is not a zonotope.[28]

Abstract algebra

[edit]

Seventeen is the highest dimension for paracompact Vineberg polytopes with rank mirror facets, with the lowest belonging to the third.[29]

17 is a supersingular prime, because it divides the order of the Monster group.[30] If the Tits group is included as a non-strict group of Lie type, then there are seventeen total classes of Lie groups that are simultaneously finite and simple (see classification of finite simple groups). In base ten, (17, 71) form the seventh permutation class of permutable primes.[31]

Other notable properties

[edit]

In science

[edit]
The elementary particles in the Standard Model of physics

Physics

[edit]

Seventeen is the number of elementary particles with unique names in the Standard Model of physics.[33]

Chemistry

[edit]

Group 17 of the periodic table is called the halogens. The atomic number of chlorine is 17.

Biology

[edit]

Some species of cicadas have a life cycle of 17 years (i.e. they are buried in the ground for 17 years between every mating season).

In religion

[edit]

Other fields

[edit]

Seventeen is:

  • The total number of syllables in a haiku (5 + 7 + 5).
  • The maximum number of strokes of a Chinese radical.

Music

[edit]

Where Pythagoreans saw 17 in between 16 from its Epogdoon of 18 in distaste,[34] the ratio 18:17 was a popular approximation for the equal tempered semitone (12-tone) during the Renaissance.

Notes

[edit]

References

[edit]
  1. ^ "random numbers". catb.org/.
  2. ^ "The Power of 17". Cosmic Variance. Archived from the original on 2008-12-04. Retrieved 2010-06-14.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A094133 (Leyland numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A094133 (Leyland prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A045575 (Leyland numbers of the second kind)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A123206 (Leyland prime numbers of the second kind)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A014556 (Euler's "Lucky" numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
  8. ^ John H. Conway and Richard K. Guy, The Book of Numbers. New York: Copernicus (1996): 11. "Carl Friedrich Gauss (1777–1855) showed that two regular "heptadecagons" (17-sided polygons) could be constructed with ruler and compasses."
  9. ^ Pappas, Theoni, Mathematical Snippets, 2008, p. 42.
  10. ^ McGuire, Gary (2012). "There is no 16-clue sudoku: solving the sudoku minimum number of clues problem". arXiv:1201.0749 [cs.DS].
  11. ^ McGuire, Gary; Tugemann, Bastian; Civario, Gilles (2014). "There is no 16-clue sudoku: Solving the sudoku minimum number of clues problem via hitting set enumeration". Experimental Mathematics. 23 (2): 190–217. doi:10.1080/10586458.2013.870056. S2CID 8973439.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A006227 (Number of n-dimensional space groups (including enantiomorphs))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
  13. ^ Dallas, Elmslie William (1855), The Elements of Plane Practical Geometry, Etc, John W. Parker & Son, p. 134.
  14. ^ "Shield - a 3.7.42 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
  15. ^ "Dancer - a 3.8.24 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
  16. ^ "Art - a 3.9.18 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
  17. ^ "Fighters - a 3.10.15 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
  18. ^ "Compass - a 4.5.20 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
  19. ^ "Broken roses - three 5.5.10 tilings". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
  20. ^ "Pentagon-Decagon Packing". American Mathematical Society. AMS. Retrieved 2022-03-07.
  21. ^ Sloane, N. J. A. (ed.). "Sequence A003323 (Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
  22. ^ Babbitt, Frank Cole (1936). Plutarch's Moralia. Vol. V. Loeb.
  23. ^ Sloane, N. J. A. (ed.). "Sequence A072895 (Least k for the Theodorus spiral to complete n revolutions)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-06-19.
  24. ^ a b Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 2022-11-26. Retrieved 2022-11-25.
  25. ^ H. S. M. Coxeter; P. Du Val; H. T. Flather; J. F. Petrie (1982). The Fifty-Nine Icosahedra. New York: Springer. doi:10.1007/978-1-4613-8216-4. ISBN 978-1-4613-8216-4.
  26. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-17.
  27. ^ Sloane, N. J. A. (ed.). "Sequence A007504 (Sum of the first n primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-17.
  28. ^ Senechal, Marjorie; Galiulin, R. V. (1984). "An introduction to the theory of figures: the geometry of E. S. Fedorov". Structural Topology (in English and French) (10): 5–22. hdl:2099/1195. MR 0768703.
  29. ^ Tumarkin, P.V. (May 2004). "Hyperbolic Coxeter N-Polytopes with n+2 Facets". Mathematical Notes. 75 (5/6): 848–854. arXiv:math/0301133. doi:10.1023/B:MATN.0000030993.74338.dd. Retrieved 18 March 2022.
  30. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
  31. ^ Sloane, N. J. A. (ed.). "Sequence A258706 (Absolute primes: every permutation of digits is a prime. Only the smallest representative of each permutation class is shown.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-29.
  32. ^ Berlekamp, E. R.; Graham, R. L. (1970). "Irregularities in the distributions of finite sequences". Journal of Number Theory. 2 (2): 152–161. Bibcode:1970JNT.....2..152B. doi:10.1016/0022-314X(70)90015-6. MR 0269605.
  33. ^ Glenn Elert (2021). "The Standard Model". The Physics Hypertextbook.
  34. ^ Plutarch, Moralia (1936). Isis and Osiris (Part 3 of 5). Loeb Classical Library edition.
[edit]
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