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

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

Seventeen is the seventh prime number, which makes it the fourth super-prime,[1] as seven is itself prime. It forms a twin prime with 19,[2] a cousin prime with 13,[3] and a sexy prime with both 11 and 23.[4] Seventeen is the only prime number which is the sum of four consecutive primes (2, 3, 5, and 7), as any other four consecutive primes that are added always generate an even number divisible by two. It is one of six lucky numbers of Euler which produce primes of the form ${\displaystyle k^{2}-k+41}$,[5] and the sixth Mersenne prime exponent, which yields 131,071.[6] It is also the minimum possible number of givens for a sudoku puzzle with a unique solution.[7][8] 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:[9][10]

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

The number of integer partitions of 17 into prime parts is 17 (the only number ${\displaystyle n}$ such that its number of such partitions is ${\displaystyle n}$).[11]

Seventeen is the third Fermat prime, as it is of the form ${\displaystyle 2^{2^{n))+1}$ with ${\displaystyle n=2}$.[12] On the other hand, the seventeenth Jacobsthal–Lucas number — that is part of a sequence which includes four Fermat primes (except for 3) — is the fifth and largest known Fermat prime: 65,537.[13] It is one more than the smallest number with exactly seventeen divisors, 65,536 = 216.[14] 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.[15][16]

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".[17]

Seventeen is the minimum number of vertices on a graph such that, if the edges are colored with three different colors, there is bound to be a monochromatic triangle; see Ramsey's theorem.[18]

There are also:

• Seventeen distinct fully supported stellations generated by an icosahedron.[28] The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules.[29][30] 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).[31]
Seventeen distinct fully supported stellations are also produced by truncated cube and truncated octahedron.[28]

Seventeen is the highest dimension for paracompact Vineberg polytopes with rank ${\displaystyle n+2}$ mirror facets, with the lowest belonging to the third.[33]

Seventeen is the longest sequence for which a solution exists in the irregularity of distributions problem,[34] while the sequence of residues (mod n) of a googol and googolplex, for ${\displaystyle n=1,2,3,...}$, agree up until ${\displaystyle n=17}$.

In abstract algebra, 17 is the seventh supersingular prime that divides the order of six sporadic groups (J3, He, Fi23, Fi24, B, and F1) inside the Happy Family of such groups.[35] The 16th and 18th prime numbers (53 and 61) are the only two primes less than 71 that do not divide the order of any sporadic group including the pariahs, with this prime as the largest such supersingular prime that divides the largest of these groups (F1). On the other hand, 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.[36]

A positive definite quadratic integer matrix represents all primes when it contains at least the set of seventeen numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}; only four prime numbers less than the largest member are not part of the set (53, 59, 61, and 71).[37]

## 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.

## 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,[39] and some jurisdictions around the world, 17 is the age of sexual consent[40]
• 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

### Music

 Main article: 17 (disambiguation) § Music

### 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.[41]
• 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:

• 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 to call police in France.
• Force 17, a special operations unit of the Palestinian Fatah movement.
• The number of the French department Charente-Maritime.
• Malaysia Airlines Flight 17 was shot down by Russian-controlled forces on 17 July 2014 after flying over eastern Ukraine. The first test flight of the plane, a Boeing 777-200ER, was on 17 July 1997, exactly 17 years prior to the doomed flight.

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-29.
2. ^ Sloane, N. J. A. (ed.). "Sequence A001359 (Lesser of twin primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
3. ^ 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.
4. ^ 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.
5. ^ Sloane, N. J. A. (ed.). "Sequence A014556 (Euler's "Lucky" numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
6. ^ Sloane, N. J. A. (ed.). "Sequence A000043 (Mersenne exponents)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
7. ^ McGuire, Gary (2012). "There is no 16-clue sudoku: solving the sudoku minimum number of clues problem". arXiv:1201.0749 [cs.DS].
8. ^ 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.
9. ^ Sloane, N. J. A. (ed.). "Sequence A094133 (Leyland primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
10. ^ Sloane, N. J. A. (ed.). "Sequence A045575 (Leyland primes of the second kind)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
11. ^ Sloane, N. J. A. (ed.). "Sequence A000607 (Number of partitions of n into prime parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-12.
12. ^ "Sloane's A019434 : Fermat primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
13. ^ Sloane, N. J. A. (ed.). "Sequence A014551 (Jacobsthal-Lucas numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-29.
14. ^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-28.
15. ^ 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."
16. ^ Pappas, Theoni, Mathematical Snippets, 2008, p. 42.
17. ^ Babbitt, Frank Cole (1936). Plutarch's Moralia. Vol. V. Loeb.
18. ^ 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.
19. ^ 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.
20. ^ Dallas, Elmslie William (1855), The Elements of Plane Practical Geometry, Etc, John W. Parker & Son, p. 134.
21. ^ "Shield - a 3.7.42 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
22. ^ "Dancer - a 3.8.24 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
23. ^ "Art - a 3.9.18 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
24. ^ "Fighters - a 3.10.15 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
25. ^ "Compass - a 4.5.20 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
26. ^ "Broken roses - three 5.5.10 tilings". Kevin Jardine's projects. Kevin Jardine. Retrieved 2022-03-07.
27. ^ "Pentagon-Decagon Packing". American Mathematical Society. AMS. Retrieved 2022-03-07.
28. ^ a b Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 2022-11-26. Retrieved 2022-11-25.
29. ^ 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.
30. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-17.
31. ^ 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.
32. ^ 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.
33. ^ 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.
34. ^
35. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
36. ^ 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.
37. ^
38. ^ Glenn Elert (2021). "The Standard Model". The Physics Hypertextbook.
39. ^ "Age Of Consent By State". Archived from the original on 2011-04-17.
40. ^ "Age of consent for sexual intercourse". 2015-06-23.
41. ^ Plutarch, Moralia (1936). Isis and Osiris (Part 3 of 5). Loeb Classical Library edition.
42. ^ "random numbers". catb.org/.
43. ^ "The Power of 17". Cosmic Variance. Archived from the original on 2008-12-04. Retrieved 2010-06-14.