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# 17 (number)

 ← 16 17 18 →
Cardinalseventeen
Ordinal17th
(seventeenth)
Numeral systemseptendecimal
Factorizationprime
Prime7th
Divisors1, 17
Greek numeralΙΖ´
Roman numeralXVII
Binary100012
Ternary1223
Quaternary1014
Quinary325
Senary256
Octal218
Duodecimal1512
VigesimalH20
Base 36H36

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.

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## In mathematics

Seventeen is the seventh prime number. The next prime is nineteen, with which it forms a twin prime.

Seventeen is a permutable prime and a supersingular prime.

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 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.

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".[4]

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.[5][6]

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 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 ((1,4) or (2,3)). The next such number is 65.

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.)

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 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,[8] and some jurisdictions around the world, 17 is the age of sexual consent[9]
• 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
• 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 culture

### Religion

• According to Leon Kass, 17 has some significant meaning (as yet not known exactly) in the book of Genesis.[10]
• 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.[11]
• 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'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. ^ Babbitt, Frank Cole (1936). Plutarch's Moralia. V. Loeb.
5. ^ McGuire, Gary (2012). "There is no 16-clue sudoku: solving the sudoku minimum number of clues problem". arXiv:1201.0749 [cs.DS].
6. ^ 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.
7. ^ Glenn Elert. "The Standard Model".
8. ^ "Age Of Consent By State". Archived from the original on 2011-04-17.
9. ^
10. ^ For example, the patriarch Jacob lived 17 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.
11. ^ Plutarch, Moralia (1936). Isis and Osiris (Part 3 of 5). Loeb Classical Library edition.
12. ^ "random numbers". catb.org/.
13. ^ "The Power of 17". Cosmic Variance.
14. ^ Ratliff, Ben (7 August 2017). "Why Would You Go to a Phish Concert, Let Alone 13? I Found Out". Retrieved 30 April 2018 – via NYTimes.com.
15. ^ "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". livemusicblog.com. 27 December 2017. Retrieved 30 April 2018.
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