17 (number)

17 is a Leyland number{{Cite OEIS|A094133|Leyland numbers}} and Leyland prime,{{Cite OEIS|A094133|Leyland prime numbers}} using 2 & 3 (2³ + 3²) and using 4 and 5,{{Cite OEIS|A045575|Leyland numbers of the second kind}}{{Cite OEIS|A123206}} using 3 & 4 (3⁴ - 4³). 17 is a Fermat prime. 17 is one of six lucky numbers of Euler.{{Cite OEIS|A014556|Euler's "Lucky" numbers|access-date=2022-11-25}}

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.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."Pappas, Theoni, Mathematical Snippets, 2008, p. 42.

The minimum possible number of givens for a sudoku puzzle with a unique solution is 17.{{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}}{{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}}

File:Wheel of Theodorus.png, with a maximum right triangles laid edge-to-edge before one revolution is completed. The largest triangle has a hypotenuse of $\backslash sqrt\; \{17\}.$]]

• There are seventeen crystallographic space groups in two dimensions.{{Cite OEIS |A006227 |Number of n-dimensional space groups (including enantiomorphs) |access-date=2022-11-25 }} These are sometimes called wallpaper groups, 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 fill a plane vertex.{{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}}. Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42,{{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}} 3.8.24,{{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}} 3.9.18,{{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}} 3.10.15,{{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}} 4.5.20,{{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}} and 5.5.10){{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}} exclusively surround a point in the plane and fill it only when irregular polygons are included.{{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}}
• Seventeen is the minimum number of vertices on a two-dimensional graph such that, if the edges are colored with three different colors, there is bound to be a monochromatic triangle; see Ramsey's theorem.{{Cite OEIS |A003323 |Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's. |access-date=2022-11-25 }}
• 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".{{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}}

17 is the least $k$ for the Theodorus Spiral to complete one revolution.{{Cite OEIS |A072895 |Least k for the Theodorus spiral to complete n revolutions |access-date=2024-06-19 }} This, in the sense of Plato, who questioned why Theodorus (his tutor) stopped at $\backslash sqrt\{17\}$ when illustrating adjacent right triangles whose bases are units and heights are successive square roots, starting with $1$. 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.

In three-dimensional space, there are seventeen distinct fully supported stellations generated by an icosahedron.{{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 }} The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules.{{Cite book |author1=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 }}{{Cite OEIS |A000040 |The prime numbers |access-date=2023-02-17 }} 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).{{Cite OEIS|A007504 |Sum of the first n primes. |access-date=2023-02-17 }} Seventeen distinct fully supported stellations are also produced by truncated cube and truncated octahedron.

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.{{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}}

Seventeen is the highest dimension for paracompact Vineberg polytopes with rank $n+2$ mirror facets, with the lowest belonging to the third.{{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}}

17 is a supersingular prime, because it divides the order of the Monster group.{{Cite OEIS |A002267 |The 15 supersingular primes |access-date=2022-11-25 }} 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.{{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 }}

• The sequence of residues (mod {{mvar|n}}) of a googol and googolplex, for $n=1,\; 2,\; 3,\; ...$, agree up until $n=17$.{{citation needed|date=August 2024}}
• Seventeen is the longest sequence for which a solution exists in the irregularity of distributions problem.{{cite journal|author1=Berlekamp, E. R. |author2=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}}

File:Standard Model of Elementary Particles.svgs in the Standard Model of physics.]]

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

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• {{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}}

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{{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}}

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Category:Integers