17th Breakthrough Artillery Division

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

17 is the seventh prime number, which makes seventeen the fourth super-prime, as seven is itself prime. 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 an emirp, and more specifically a permutable prime with 71, both of which are also supersingular primes. Seventeen is the sixth Mersenne prime exponent, yielding 131,071.

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.

Seventeen can be written in the form $x^y + y^x$ and $x^y - y^x$, and, as such, it is a Leyland prime and Leyland prime of the second kind:
:$17=2^+3^=3^-4^$.

17 is one of seven lucky numbers of Euler which produce primes of the form $k^-k+41$.

Seventeen is the third Fermat prime, as it is of the form $2^ + 1$, specifically with $n=2$. Since 17 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.

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".

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.

There are also:

*17 crystallographic space groups in two dimensions. These are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper.

*17 combinations of regular polygons that completely fill a plane vertex. Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, and 5.5.10) exclusively surround a point in the plane and fill it only when irregular polygons are included.

*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.

*17 distinct fully supported stellations generated by an icosahedron. Inclusive of the icosahedron, there are 59 total stellations by Miller's rules. The truncated cube and truncated octahedron also produce 17 distinct fully supported stellations.

*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.

Seventeen is the highest dimension for paracompact Vinberg polytopes with rank $n+2$ mirror facets, with the lowest belonging to the third.

Seventeen is the minimum possible number of givens for a sudoku puzzle with a unique solution. This was long conjectured, and was proved between 2012 and 2014.

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

A positive definite quadratic integer matrix represents all primes when it contains at least the set of 17 numbers: . Only four prime numbers up to 73 are not part of the set.

In science

* The atomic number of chlorine.
* The Brodmann area defining the primary visual processing area of mammalian brains.
* Group 17 of the periodic table is called the halogens.
* The number of elementary particles with unique names in the Standard Model of physics.

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, and some jurisdictions around the world, 17 is the age of sexual consent
* 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

Bands

* 17 Hippies, a German band
* Seventeen (), a South Korean boy band
* Heaven 17, an English new wave band
* East 17, an English boy band

Albums

* ''17'' (XXXTentacion album)
* ''17'' (Motel album)
* ''17'' (Ricky Martin album)
* ''Chicago 17'', a 1984 album by Chicago
* ''Seventeen Days'', a 2005 album by 3 Doors Down
* ''Seventeen Seconds'', a 1980 album by The Cure
* ''17 Carat'', a 2015 EP by Seventeen

Songs

* "17 Again", a song by Tide Lines
* "17" (Sky Ferreira song)
* "17" (Yourcodenameis:Milo song)
* " 17 Again", a song by Eurythmics
* "17 år", a song by Veronica Maggio
* "17 Crimes", a song by AFI
* " 17 Days", a song by Prince
* "17", a song by Dan Bălan
* "17", a song by Jethro Tull
* "17", a song by Kings of Leon
* "17", a song by Milburn
* "17", a song by Rick James from '' Reflections''
* "17", a B-side by Shiina Ringo on the "Tsumi to Batsu" single
* "17", a song by The Smashing Pumpkins from the album '' Adore''
* "17", a song by Youth Lagoon from the album ''The Year of Hibernation''
* "17 Days", a song by Prince & the Revolution, B side from the 1984 "When Doves Cry" single
* "Seventeen" (Jet song)
* "Seventeen" (Ladytron song)
* "Seventeen" (Winger song)
* "Seventeen", a song by ¡Forward, Russia! from ''Give Me a Wall''
* "Seventeen", a song by Jimmy Eat World from ''Static Prevails''
* "Seventeen", a song by Marina & the Diamonds from the US edition of '' The Family Jewels''
* "Seventeen", a song by Mat Kearney from the iTunes edition of '' Young Love''
* "Seventeen", a song from the ''Repo! The Genetic Opera'' soundtrack
* "Seventeen", the original title of the song " I Saw Her Standing There" by The Beatles
* "Seventeen", a song by the Sex Pistols from ''Never Mind the Bollocks, Here's the Sex Pistols''
* "Seventeen Forever", a song by Metro Station
* "At Seventeen", a song by Janis Ian
* " Edge of Seventeen", a song by Stevie Nicks
* "Seventeen Ain't So Sweet", a song by The Red Jumpsuit Apparatus from ''Don't You Fake It''
* "Only 17", a song by Rucka Rucka Ali
* "Opus 17 (Don't You Worry 'Bout Me)", a song by Frankie Valli and the Four Seasons
* "(She's) Sexy + 17", a song by Stray Cats from '' Rant N' Rave with the Stray Cats''
* "Hello, Seventeen", a song by 12012
* "Section 17 (Suitcase Calling)", a song by The Polyphonic Spree
* "Day Seventeen: Accident?", a song by Ayreon
* "Seventeen", a song by Alessia Cara
* "Seventeen", a song performed by Marina and the Diamonds
* "Seventeen" and "Seventeen (Reprise)", songs in the musical ''Heathers''
* "Seventeen" and "Seventeen (Reprise)", songs in the musical ''Tuck Everlasting''

Other

* ''Seventeen'', a 1951 American musical
* The ratio 18:17 was a popular approximation for the equal tempered semitone during the Renaissance

Film

* ''Seventeen'' (1916), an adaptation of the novel of the same name by Booth Tarkington
* '' Number 17'' (1928), a British-German film
* ''Number Seventeen'' (1932), directed by Alfred Hitchcock
* ''Seventeen'' (1940), a second adaptation of the Tarkington novel
* '' Number 17'' (1949), a Swedish film
* '' Stalag 17'' (1953), directed by Billy Wilder
* ''Try Seventeen'' (2002), directed by Jeffrey Porter
*'' 17 Again'' (2009), directed by Burr Steers

Anime and manga

* Android 17, a character from the ''Dragon Ball'' series
* Detective Konawaka from the ''Paprika'' anime has a strong dislike for the number 17

Games

* The computer game '' Half-Life 2'' takes place in and around City 17
* The visual novel '' Ever 17: The Out of Infinity'' 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 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.
* In the Yasna