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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.

⚫
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>{{efn|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.<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> }}


== 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 ===
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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.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>
*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>


*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>
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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 ====
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>


== In science==
== In science==
[[File:Standard Model of Elementary Particles.svg|right|240px|thumb|The [[elementary particle]]s in the [[Standard Model|Standard Model]] of physics ]]
[[File:Standard Model of Elementary Particles.svg|right|240px|thumb|The [[elementary particle]]s in the [[Standard Model]] of physics ]]


=== 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 [[Particle physics|particles]] that are either their own [[antiparticle]] (i.e. from four of five classes of [[Boson#Elementary bosons|scalar or vector bosons]]) and those that are not (collectively, twelve [[quark]]s and [[lepton]]s, and the [[W and Z boson|W boson]]); that is, without distinguishing between eight [[gluon]]s, or two W<sup>±</sup> bosons (as each other's antiparticles, W<sup>+</sup> and W<sup>−</sup>).


=== Chemistry ===
=== Chemistry ===
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== In religion ==
== In religion ==
* In the [[Yasna]] of [[Zoroastrianism]], seventeen chapters were written by [[Zoroaster]] himself. These are the five [[Gathas]].
* In the [[Yasna]] of [[Zoroastrianism]], seventeen chapters were written by [[Zoroaster]] himself. These are the five [[Gathas]].
* The number of [[sura]]t [[al-Isra]] in the [[Qur'an]] is seventeen, as is the total number of [[Rak'a#Daily prayers|Rak'a]]s that Muslims perform during [[Salat]] on a daily basis.
* 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 ==
== Other fields ==
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=== Music ===
=== Music ===


Where [[Pythagorean]]s saw 17 in between 16 from its [[Epogdoon]] of 18 in distaste, the ratio 18:17 was a popular approximation for the [[equal temperament|equal tempered]] [[semitone]] during the Renaissance.<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>
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]].


== Notes ==
== Notes ==
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{{wiktionary|seventeen}}
{{wiktionary|seventeen}}


* [https://primes.utm.edu/curios/page.php/17.html Prime Curios for the number 17]
* [https://primes.utm.edu/curios/page.php/17.html Prime Curios!: 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?]


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

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

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

Two-dimensions

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

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

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

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

In science

The elementary particles in the Standard Model of physics

Physics

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

Chemistry

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

Biology

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

Other fields

Seventeen is:

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

Music

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

References

  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.
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