17 (number)
Template:Other uses Template:Infobox number 17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.
Mathematics
17 is a Leyland number<ref>Template:Cite OEIS</ref> and Leyland prime,<ref>Template:Cite OEIS</ref> using 2 & 3 (23 + 32),<ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref> and using 3 & 4 (34 - 43). 17 is a Fermat prime. 17 is one of six lucky numbers of Euler.<ref>Template:Cite OEIS</ref>
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.<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>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>Template:Cite arXiv</ref><ref>Template:Cite journal</ref>
Geometric properties
Two-dimensions
- There are seventeen crystallographic space groups in two dimensions.<ref>Template:Cite OEIS</ref> 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.<ref>Template:Citation.</ref> Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42,<ref>Template:Cite web</ref> 3.8.24,<ref>Template:Cite web</ref> 3.9.18,<ref>Template:Cite web</ref> 3.10.15,<ref>Template:Cite web</ref> 4.5.20,<ref>Template:Cite web</ref> and 5.5.10)<ref>Template:Cite web</ref> exclusively surround a point in the plane and fill it only when irregular polygons are included.<ref>Template:Cite web</ref>
- 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.<ref>Template:Cite OEIS</ref>
- 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".<ref>Template:Cite book</ref>
17 is the least <math>k</math> for the Theodorus Spiral to complete one revolution.<ref>Template:Cite OEIS</ref> This, in the sense of Plato, who questioned why Theodorus (his tutor) stopped at <math>\sqrt{17}</math> when illustrating adjacent right triangles whose bases are units and heights are successive square roots, starting with <math>1</math>. 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.<ref name="Stellations">Template:Cite web</ref> The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules.<ref>Template:Cite book</ref><ref>Template:Cite OEIS</ref> 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).<ref>Template:Cite OEIS</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 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.<ref>Template:Cite journal</ref>
Abstract algebra
Seventeen is the highest dimension for paracompact Vineberg polytopes with rank <math>n+2</math> mirror facets, with the lowest belonging to the third.<ref>Template:Cite journal</ref>
17 is a supersingular prime, because it divides the order of the Monster group.<ref>Template:Cite OEIS</ref> 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.<ref>Template:Cite OEIS</ref>
Other notable properties
- The sequence of residues (mod Template:Mvar) of a googol and googolplex, for <math>n=1, 2, 3, ...</math>, agree up until <math>n=17</math>.Template:Citation needed
- Seventeen is the longest sequence for which a solution exists in the irregularity of distributions problem.<ref>Template:Cite journal</ref>
Other fields
Music
Where Pythagoreans saw 17 in between 16 from its Epogdoon of 18 in distaste,<ref>Template:Cite book</ref> the ratio 18:17 was a popular approximation for the equal tempered semitone (12-tone) during the Renaissance.
Art
17 is the number of syllables in a traditional Japanese haiku, arranged in 3 lines of 5, 7, and 5 syllables.
Psychology
17 was described at MIT as "the least random number", according to the Jargon File.<ref>Template:Cite web</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>Template:Cite web</ref>
Notes
References
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