17 (number)

 ← 16 17 18 →
Cardinalseventeen
Ordinal17th
(seventeenth)
Numeral systemseptendecimal
Factorizationprime
Prime7th
Divisors1, 17
Greek numeralΙΖ´
Roman numeralXVII
Binary100012
Ternary1223
Senary256
Octal218
Duodecimal1512

17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.

Seventeen is the sum of the first four prime numbers.

In mathematics

17 is the seventh prime number, which makes seventeen the fourth super-prime, as seven is itself prime. The next prime is 19, with which it forms a twin prime.[1] It is a cousin prime with 13 and a sexy prime with 11 and 23.[2][3] It is an emirp, and more specifically a permutable prime with 71, both of which are also supersingular primes.[4][5]

Seventeen is the only prime number which is the sum of four consecutive primes: 2,3,5,7. Any other four consecutive primes summed would always produce an even number, thereby divisible by 2 and so not prime.

Seventeen can be written in the form ${\displaystyle x^{y}+y^{x}}$ and ${\displaystyle x^{y}-y^{x}}$, and, as such, it is a Leyland prime and Leyland prime of the second kind:[6][7]

${\displaystyle 17=2^{3}+3^{2}=3^{4}-4^{3}}$.

17 is one of seven lucky numbers of Euler which produce primes of the form ${\displaystyle k^{2}-k+41}$.[8]

Seventeen is the sixth Mersenne prime exponent, yielding 131,071.[9]

Seventeen is the third Fermat prime, as it is of the form ${\displaystyle 2^{2^{n}}+1}$, specifically with ${\displaystyle n=2}$.[10] Since 17 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.[11][12]

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".[13]

Seventeen is the minimum number of vertices on a graph such that, if the edges are coloured with three different colours, there is bound to be a monochromatic triangle; see Ramsey's theorem.[14]

There are also:

• 17 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 seventeen prime numbers.[27]
17 distinct fully supported stellations are also produced by truncated cube and truncated octahedron.[24]

Seventeen is the highest dimension for paracompact Vinberg polytopes with rank ${\displaystyle n+2}$ mirror facets, with the lowest belonging to the third.[29]

Seventeen is the minimum possible number of givens for a sudoku puzzle with a unique solution. This was long conjectured, and was proved between 2012 and 2014.[30][31]

The sequence of residues (mod n) of a googol and googolplex, for ${\displaystyle n=1,2,3,...}$, agree up until ${\displaystyle n=17}$.

A positive definite quadratic integer matrix represents all primes when it contains at least the set of 17 numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}.[32] Only four prime numbers up to 73 are not part of the set.

In languages

Grammar

In Catalan, 17 is the first compound number (disset). The numbers 11 (onze) through 16 (setze) have their own names.

In French, 17 is the first compound number (dix-sept). The numbers 11 (onze) through 16 (seize) have their own names.

In Italian, 17 is also the first compound number (diciassette), whereas sixteen is sedici.

Age 17

• In most countries across the world, it is the last age at which one is considered a minor under law.
• In the UK, the minimum age for taking driving lessons, and to drive a car or a van
• In the US and Canada, it is the age at which one may purchase, rent, or reserve M-rated video games without parental consent
• In some US states,[34] and some jurisdictions around the world, 17 is the age of sexual consent[35]
• In most US states, Canada and in the UK, the age at which one may donate blood (without parental consent)
• In many countries and jurisdictions, the age at which one may obtain a driver's license
• In the US, the age at which one may watch, rent, or purchase R-rated movies without parental consent
• The U.S. TV Parental Guidelines system sets 17 as the minimum age one can watch programs with a TV-MA rating without parental guidance.
• In the US, the age at which one can enlist in the armed forces with parental consent
• In the US, the age at which one can apply for a private pilot licence for powered flight (however, applicants can obtain a student pilot certificate at age 16)
• In Greece and Indonesia, the voting age
• In Chile and Indonesia, the minimum driving age.
• In Tajikistan, North Korea and Timor-Leste, the age of majority

In culture

Print

• The title of Seventeen, a magazine
• The title of Just Seventeen, a former magazine
• The number 17 is a recurring theme in the works of novelist Steven Brust. All of his chaptered novels have either 17 chapters or two books of 17 chapters each. Multiples of 17 frequently appear in his novels set in the fantasy world of Dragaera, where the number is considered holy.
• In The Illuminatus! Trilogy, the symbol for Discordianism includes a pyramid with 17 steps because 17 has "virtually no interesting geometric, arithmetic, or mystical qualities". However, for the Illuminati, 17 is tied with the "23/17 phenomenon".
• In the Harry Potter universe
• 17 is the coming of age for wizards. It is equivalent to the usual coming of age at 18.
• 17 is the number of Sickles in one Galleon in the British wizards' currency.

Religion

• According to Plutarch's Moralia, the Egyptians have a legend that the end of Osiris' life came on the seventeenth of a month, on which day it is quite evident to the eye that the period of the full moon is over. Now, because of this, the Pythagoreans call this day "the Barrier", and utterly abominate this number. For the number seventeen, coming in between the square sixteen and the oblong rectangle eighteen, which, as it happens, are the only plane figures that have their perimeters equal their areas, bars them off from each other and disjoins them, and breaks up the epogdoon by its division into unequal intervals.[36]
• In the Yasna of Zoroastrianism, seventeen chapters were written by Zoroaster himself. These are the Gathas.
• The number of the raka'ahs that Muslims perform during Salat on a daily basis.
• The number of surat al-Isra in the Qur'an.

In other fields

Seventeen is:

No row 17 in Alitalia planes

References

1. ^ Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
2. ^ Sloane, N. J. A. (ed.). "Sequence A046132 (Larger member p+4 of cousin primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
3. ^ Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
4. ^ Sloane, N. J. A. (ed.). "Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
5. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
6. ^ Sloane, N. J. A. (ed.). "Sequence A094133 (Leyland primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
7. ^ Sloane, N. J. A. (ed.). "Sequence A045575". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25. Leyland primes of the second kind.
8. ^ Sloane, N. J. A. (ed.). "Sequence A014556 (Euler's "Lucky" numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
9. ^ Sloane, N. J. A. (ed.). "Sequence A000043 (Mersenne exponents)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
10. ^ "Sloane's A019434 : Fermat primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. ^ 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."
12. ^ Pappas, Theoni, Mathematical Snippets, 2008, p. 42.
13. ^ Babbitt, Frank Cole (1936). Plutarch's Moralia. Vol. V. Loeb.
14. ^ 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.
15. ^ 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.
16. ^ Dallas, Elmslie William (1855), The Elements of Plane Practical Geometry, Etc, John W. Parker & Son, p. 134.
17. ^ "Shield - a 3.7.42 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
18. ^ "Dancer - a 3.8.24 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
19. ^ "Art - a 3.9.18 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
20. ^ "Fighters - a 3.10.15 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
21. ^ "Compass - a 4.5.20 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
22. ^ "Broken roses - three 5.5.10 tilings". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
23. ^ "Pentagon-Decagon Packing". American Mathematical Society. AMS. Retrieved 2022-03-07.
24. ^ a b Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 2022-11-25. 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. ^ McGuire, Gary (2012). "There is no 16-clue sudoku: solving the sudoku minimum number of clues problem". arXiv:1201.0749 [cs.DS].
31. ^ 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.
32. ^
33. ^ Glenn Elert (2021). "The Standard Model". The Physics Hypertextbook.
34. ^ "Age Of Consent By State". Archived from the original on 2011-04-17.
35. ^ "Age of consent for sexual intercourse". 2015-06-23.
36. ^ Plutarch, Moralia (1936). Isis and Osiris (Part 3 of 5). Loeb Classical Library edition.
37. ^ "random numbers". catb.org/.
38. ^ "The Power of 17". Cosmic Variance.
39. ^ Ratliff, Ben (7 August 2017). "Why Would You Go to a Phish Concert, Let Alone 13? I Found Out". The New York Times. Retrieved 30 April 2018.
40. ^ "Phish Returns to Madison Square Garden for New Year's Eve; Here's What We Think Will Go Down (Hint: Cosmic Wristbands) - LIVE music blog". Live Music Blog. 27 December 2017. Retrieved 30 April 2018.