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17 (number)

## Summary

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

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

Seventeen is the sum of the first four prime numbers.

## In mathematics

17 is the seventh prime number. The next prime is 19, with which it forms a twin prime. It is a cousin prime with 13, and a sexy prime with 11 and 23. It is a permutable prime with 71, and a supersingular prime (as is 71, the largest supersingular prime). 17 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: 17 = 23 + 32 = 34 - 43.

Seventeen is the third Fermat prime, as it is of the form 22n + 1, specifically with n = 2.[1] Since 17 is a Fermat prime, regular heptadecagons can be constructed with a compass and unmarked ruler. This was proven by Carl Friedrich Gauss[2] and ultimately led him to choose mathematics over philology for his studies.[3]

There are 17 two-dimensional space (plane symmetry) groups. These are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper.

There are 17 two-dimensional combinations of regular polygons that completely fill a plane vertex.[4] 11 of these belong to regular and semiregular tilings, while 6 of these (3.7.42,[5] 3.8.24,[6] 3.9.18,[7] 3.10.15,[8] 4.5.20,[9] and 5.5.10)[10] exclusively surround a point in the plane, and fill it only when irregular polygons are included.[11]

There are 17 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.[12]

Seventeen is the highest dimension for paracompact Vinberg polytopes of rank n+2, with the lowest belonging to the third.[13]

Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, for all positive n < p − 1.

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".[14]

Seventeen is the minimum possible number of givens for a sudoku puzzle with a unique solution. This was long conjectured, and was proved in 2012–14.[15][16]

There are 17 orthogonal curvilinear coordinate systems (to within a conformal symmetry) in which the three-variable Laplace equation can be solved using the separation of variables technique.

Seventeen is the sixth Mersenne prime exponent, yielding 131071.

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.)

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}.[17] Only four prime numbers up to 73 are not part of the set.

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.

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

## 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,[19] and some jurisdictions around the world, 17 is the age of sexual consent[20]
• 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, it is 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.[21]
• 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
• In Italian culture, the number 17 is considered unlucky. When viewed as the Roman numeral, XVII, it is then changed anagrammatically to VIXI, which in the Latin language translates to "I lived", the perfect implying "My life is over." (c.f. "Vixerunt", Cicero's famous announcement of an execution.) Renault sold its "R17" model in Italy as "R177." See Cesana Pariol in the sport section about the name of curve 17.
• The fear of the number 17 is called 'heptadecaphobia' or 'heptakaidekaphobia'
• Some species of cicadas have a life cycle of 17 years (i.e. they are buried in the ground for 17 years between every mating season)
• The number of special significance to Yellow Pig's Day and Hampshire College Summer Studies in Mathematics
• The number to call police in France
• Force 17, a special operations unit of the Palestinian Fatah movement
• The number of the French department Charente-Maritime
• The declared percentage alcohol content (by volume) of Baileys Irish Cream - an Irish whiskey and cream based liqueur, made by Gilbeys of Ireland
• The flight number of Malaysia Airlines Flight 17 which was shot down on 17 July 2014 with first test flight of plane is on 17 July 1997 exactly 17 years.
• The record number of concerts performed in a single year at Madison Square Garden by the band Phish in 2017[24][25]
• The number of deaths and injured people on February 14, 2018, shooting at the Marjory Stoneman Douglas High School shooting.

## References

1. ^ "Sloane's A019434 : Fermat primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
2. ^ 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."
3. ^ Pappas, Theoni, Mathematical Snippets, 2008, p. 42.
4. ^ Dallas, Elmslie William (1855), The Elements of Plane Practical Geometry, Etc, John W. Parker & Son, p. 134.
5. ^ "Shield - a 3.7.42 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
6. ^ "Dancer - a 3.8.24 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
7. ^ "Art - a 3.9.18 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
8. ^ "Fighters - a 3.10.15 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
9. ^ "Compass - a 4.5.20 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
10. ^ "Broken roses - three 5.5.10 tilings". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
11. ^ "Pentagon-Decagon Packing". American Mathematical Society. AMS. Retrieved 2022-03-07.
12. ^ 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.
13. ^ Tumarkin, P.V. (May 2004). "Hyperbolic Coxeter N-Polytopes with n+2 Facets". Mathematical Notes. Springer. 75 (5/6): 848–854. doi:10.1023/B:MATN.0000030993.74338.dd. Retrieved 18 March 2022.
14. ^ Babbitt, Frank Cole (1936). Plutarch's Moralia. Vol. V. Loeb.
15. ^ McGuire, Gary (2012). "There is no 16-clue sudoku: solving the sudoku minimum number of clues problem". arXiv:1201.0749 [cs.DS].
16. ^ 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.
17. ^ Sloane, N. J. A. (ed.). "Sequence A154363 (Numbers from Bhargava's prime-universality criterion theorem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
18. ^ Glenn Elert (2021). "The Standard Model". The Physics Hypertextbook.
19. ^ "Age Of Consent By State". Archived from the original on 2011-04-17.
20. ^ "Age of consent for sexual intercourse". 2015-06-23.
21. ^ Plutarch, Moralia (1936). Isis and Osiris (Part 3 of 5). Loeb Classical Library edition.
22. ^ "random numbers". catb.org/.
23. ^ "The Power of 17". Cosmic Variance.
24. ^ 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.
25. ^ "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.