View source for 17 (number)

{{other uses|17 (disambiguation){{!}}17}}
{{Infobox number
| number =  17
| numeral = septendecimal
| factorization = [[prime number|prime]]
| prime = 7th
| divisor = 1, 17
|lang1=[[Hebrew numerals|Hebrew numeral]]|lang1 symbol=<span style="font-size:150%;">י"ז</span>|lang2=[[Babylonian cuneiform numerals|Babylonian numeral]]|lang2 symbol=𒌋𒐛}}
'''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>

== 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>) and using 4 and 5,<ref>{{Cite OEIS|A045575|Leyland numbers of the second kind}}</ref><ref>{{Cite OEIS|A123206}}</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>

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 (math book)|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>

=== Geometric properties ===
==== Two-dimensions ====
[[File:Wheel of Theodorus.png|upright=1.45|right|thumb|The [[Spiral of Theodorus]], with a maximum [[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]].
*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>
*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>

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 ====
In three-dimensional space, there are seventeen distinct [[Stellation#Stellating polyhedra|fully supported stellations]] generated by an [[Regular icosahedron|icosahedron]].<ref name="Stellations">{{Cite web |url=https://www.software3d.com/Enumerate.php |last=Webb |first=Robert |title=Enumeration of Stellations |website=www.software3d.com |access-date=2022-11-25 |archive-url=https://archive.today/20221126015207/https://www.software3d.com/Enumerate.php |archive-date=2022-11-26 }}</ref> The seventeenth prime number is [[59 (number)|59]], which is equal to the total number of stellations of the icosahedron by [[Stellation#Miller's rules|Miller's rules]].<ref>{{Cite book |author1=[[Coxeter|H. S. M. Coxeter]] |author2=P. Du Val |author3=H. T. Flather |author4=J. F. Petrie |title=The Fifty-Nine Icosahedra |publisher=Springer |location=New York |year=1982 |doi=10.1007/978-1-4613-8216-4 |isbn=978-1-4613-8216-4 }}</ref><ref>{{Cite OEIS |A000040 |The prime numbers |access-date=2023-02-17 }}</ref> Without counting the icosahedron as a ''zeroth'' stellation, this total becomes [[58 (number)|58]], a count equal to the sum of the first seven prime numbers (2 + 3 + 5 + 7 ... + 17).<ref>{{Cite OEIS|A007504 |Sum of the first n primes. |access-date=2023-02-17 }}</ref> Seventeen distinct fully supported stellations are also produced by [[truncated cube]] and [[truncated octahedron]].<ref name="Stellations"/>

==== Four-dimensional zonotopes ====
Seventeen is also the number of four-dimensional [[Parallelohedron|parallelotopes]] that are [[Zonohedron#Zonotopes|zonotopes]]. Another 34, or twice 17, are [[Minkowski sum]]s of zonotopes with the [[24-cell]], itself the simplest parallelotope that is not a zonotope.<ref>{{cite journal|last1=Senechal|first1=Marjorie|author1-link=Marjorie Senechal|last2=Galiulin|first2=R. V.|hdl=2099/1195|issue=10|journal=Structural Topology|language=en,fr|mr=768703|pages=5–22|title=An introduction to the theory of figures: the geometry of E. S. Fedorov|year=1984}}</ref>

==== Abstract algebra ====

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 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 ===
* 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>.{{citation needed|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>

[[File:Standard Model of Elementary Particles.svg|right|240px|thumb|There are seventeen [[elementary particle]]s in the [[Standard Model]] of physics.]]

== Other fields ==
=== Music ===
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 ==
{{Notelist}}

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

* [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?]

{{Integers|zero}}

{{DEFAULTSORT:17 (Number)}}
[[Category:Integers]]