17 (number)

For other uses, see 17.

Natural number

| ← 16  | 17 |  18 →

|
|

← 10 11 12 13 14 15 16 17 18 19 →

* List of numbers
* Integers

← 0 10 20 30 40 50 60 70 80 90 →

Cardinal| seventeen
Ordinal| 17th
(seventeenth)
Numeral system| septendecimal
Factorization| prime
Prime| 7th
Divisors| 1, 17
Greek numeral| ΙΖ´
Roman numeral| XVII, xvii
Binary| 100012
Ternary| 1223
Senary| 256
Octal| 218
Duodecimal| 1512
Hexadecimal| 1116
Hebrew numeral| י"ז
Babylonian numeral| 𒌋𒐛

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) and using 4 and 5,[5][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

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The Spiral of Theodorus, with a maximum right triangles laid edge-to-edge before one revolution is completed. The largest triangle has a hypotenuse of  17 . {\displaystyle {\sqrt {17}}.}

* There are seventeen crystallographic space groups in two dimensions.[12] 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.[13] Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42,[14] 3.8.24,[15] 3.9.18,[16] 3.10.15,[17] 4.5.20,[18] and 5.5.10)[19] exclusively surround a point in the plane and fill it only when irregular polygons are included.[20]
* 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.[21]
* 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  k {\displaystyle k} for the Theodorus Spiral to complete one revolution.[23] This, in the sense of Plato, who questioned why Theodorus (his tutor) stopped at  17 {\displaystyle {\sqrt {17}}} when illustrating adjacent right triangles whose bases are units and heights are successive square roots, starting with  1 {\displaystyle 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.

#### 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  n + 2 {\displaystyle n+2} 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]

* The sequence of residues (mod n) of a googol and googolplex, for  n = 1 , 2 , 3 , . . . {\displaystyle n=1,2,3,...} , agree up until  n = 17 {\displaystyle n=17} .[citation needed]
* Seventeen is the longest sequence for which a solution exists in the irregularity of distributions problem.[32]

There are seventeen elementary particles in the Standard Model of physics.

## Other fields

[edit]

### Music

[edit]

Where Pythagoreans saw 17 in between 16 from its Epogdoon of 18 in distaste,[33] 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". 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. ^ Plutarch, Moralia (1936). Isis and Osiris (Part 3 of 5). Loeb Classical Library edition.

* 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.

## External links

[edit]

Wikimedia Commons has media related to 17 (number).

Look up seventeen in Wiktionary, the free dictionary.

* Prime Curios!: 17
* Is 17 the "most random" number?

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