# 17 (number)

"Seventeen" redirects here. For other uses, see 17 (disambiguation).
 ← 16 17 18 →
Cardinal seventeen
Ordinal 17th
(seventeenth)
Factorization prime
Divisors 1, 17
Roman numeral XVII
Binary 100012
Ternary 1223
Quaternary 1014
Quinary 325
Senary 256
Octal 218
Duodecimal 1512
Vigesimal H20
Base 36 H36

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

In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70 . However, in dates such as 1789 or when contrasting numbers in the teens, such as 16, 17, 18, the stress shifts to the first syllable: 17 .

The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports, and other cultural phenomena.

## In mathematics

Seventeen is the 7th prime number. The next prime is nineteen, with which it forms a twin prime. 17 is the sum of the first four primes. 17 is the sixth Mersenne prime exponent, yielding 131071. 17 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1.

17 is the third Fermat prime, as it is of the form $2^{2^n}+1,$ specifically with n = 2, and it is also a Proth prime. Since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss.[1] Another consequence of 17 being a Fermat prime is that it is not a Higgs prime for squares or cubes; in fact, it is the smallest prime not to be a Higgs prime for squares, and the smallest not to be a Higgs prime for cubes.

17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime.

17 is the thirteenth term of the Euclid–Mullin sequence.

Seventeen is the aliquot sum of the semiprime 39, and is the aliquot sum of the semiprime 55, and is the base of the 17-aliquot tree.

There are exactly 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.

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

In the Irregularity of distributions problem, consider a sequence of real numbers between 0 and 1 such that the first two lie in different halves of this interval, the first three in different thirds, and so forth. The maximum possible length of such a sequence is 17 (Berlekamp & Graham, 1970, example 63).

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".[2]

17 is the tenth Perrin number, preceded in the sequence by 7, 10, 12.

In base 9, the smallest prime with a composite sum of digits is 17.

17 is known as the Feller number, after the famous mathematician William Feller who taught at Princeton University for many years. Feller would say, when discussing an unsolved mathematical problem, that if it could be proved for the case n = 17 then it could be proved for all positive integers n. He would also say in lectures, "Let's try this for an arbitrary value of n, say n = 17."[3]

Similar to Feller, Prof. Vadim Khayms of Stanford University is also known to use 17 as an arbitrary value during lectures. His Computational Mathematics for Engineers course includes 17 lectures.

17 is the least random number,[4] according to the Hackers' Jargon File.

It is a repunit prime in hexadecimal (11).

17 is the minimum possible number of givens for a sudoku puzzle with a unique solution. This was long conjectured, and was proved in 2012.[5]

There are 17 orthogonal curvilinear coordinate systems (to within a conformal symmetry) in which the 3-variable Laplace equation can be solved using the separation of variables technique.

17 is the first number that can be written as the sum of a positive cube and a positive square in two different ways; that is, the smallest n such that x3 + y2 = n has two different solutions for x and y positive integers. The next such number is 65.

17 is the minimum number of vertices on a graph such that, if the edges are coloured with 3 different colours, there is bound to be a monochromatic triangle. (See Ramsey's Theorem.)

17 is a full reptend prime in base 10, because its repeating decimal is 16 digits long.

## Age 17

• In the UK, the minimum driving age for a car or 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.
• Also, in some US states,[7] and some jurisdictions around the world, 17 is the age of sexual consent.[8]
• In most US states, Canada and in the UK, the age at which you may donate blood (without parental consent).
• In many countries and regions, 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.
• In the US, the age at which one can enlist in the armed forces with parental permission.
• At this age one can apply for a Private Pilot Licence (however the applicant can start training at 16).

## In culture

### Music

#### Bands

• 17 Hippies, a German band
• Sytten (17 in Norwegian), a Norwegian-American rapper

### Games

• The computer game Half-Life 2 takes place in and around City 17.
• The visual novel Ever17 strongly revolves around the number 17.

### 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 Leon Kass, 17 has some significant meaning (as yet not known exactly) in the book of Genesis.[9]
• According to Plutarch's Moralia, the Egyptians have a legend that the end of Osiris's 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.[10]
• 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 it translates to "I have 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.

## References

1. ^ 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 polygon) could be constructed with ruler and compasses."
2. ^ Babbitt, Frank Cole (1936). Plutarch's Moralia V. Loeb.
3. ^ Language Log: Another trip down Random Rd
4. ^ "Random numbers"
5. ^ McGuire, Gary. "There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem". arXiv:1201.0749. Retrieved 26 March 2012.
6. ^ http://physics.info/standard/
7. ^ http://www.age-of-consent.info/
8. ^ http://www.avert.org/age-of-consent.htm
9. ^ For example, the patriarch Jacob lived 17 years years after his son Joseph went missing and presumed dead, and lived 17 years after their reunion in Egypt, and the lifespans of Abraham aged 175, Isaac aged 180, and Jacob aged 147 are not a coincidence. "(The sum of the factors in all three cases is 17; of what possible significance this is, I have no idea.)" Leon Kass, The beginning of wisdom: reading Genesis,(Simon and Schuster, 2003), ISBN 978-0-7432-4299-8, p. 413 n. 10 (citing Genesis 47:28), quote from p. 629 n. 18, found at Google Books. Retrieved June 17, 2009.
10. ^ http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/Isis_and_Osiris*/C.html
11. ^ http://scienceblogs.com/cognitivedaily/2007/02/is_17_the_most_random_number.php
12. ^ http://blogs.discovermagazine.com/cosmicvariance/2007/01/30/the-power-of-17/