17 (number)

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161718
Cardinalseventeen
Ordinal17th
(seventeenth)
Numeral systemseptendecimal
Factorizationprime
Divisors1, 17
Roman numeralXVII
Binary100012
Ternary1223
Quaternary1014
Quinary325
Senary256
Octal218
Duodecimal1512
Hexadecimal1116
VigesimalH20
Base 36H36
 
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"Seventeen" redirects here. For other uses, see 17 (disambiguation).
161718
Cardinalseventeen
Ordinal17th
(seventeenth)
Numeral systemseptendecimal
Factorizationprime
Divisors1, 17
Roman numeralXVII
Binary100012
Ternary1223
Quaternary1014
Quinary325
Senary256
Octal218
Duodecimal1512
Hexadecimal1116
VigesimalH20
Base 36H36
TUCPamplona17.svg

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 /sɛvɨnˈtn/ vs 70 /ˈsɛvɨnti/. 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 /ˈsɛvɨntn/.

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[edit]

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.

In science[edit]

Age 17[edit]

In culture[edit]

Music[edit]

Bands[edit]

Albums[edit]

Songs[edit]

Other[edit]

Film[edit]

Anime and manga[edit]

Games[edit]

Print[edit]

Religion[edit]

In sports[edit]

In other fields[edit]

Seventeen is:

No row 17 in Alitalia planes.

References[edit]

  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/

External links[edit]