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17 (number)

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[edit]

Seventeen is the seventh prime number, which makes it the fourth super-prime,^[1]asseven is itself prime.

Prime properties[edit]

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 forms a twin prime with 19,^[2]acousin prime with 13,^[3] and a sexy prime with both 11 and 23.^[4] Furthermore,

• It is the sixth Mersenne prime exponent for numbers of the form ${\displaystyle 2^{n}-1}$, yielding 131071.^[5]
• It is also one of six lucky numbers of Euler ${\displaystyle n}$ which produce primes of the form ${\displaystyle m^{2}-m+n}$ for ${\displaystyle m=0,\ldots ,n-1.}$ (I.e. for ${\displaystyle n}$ of 17 and ${\displaystyle m}$ of 16 there is 257.)^[6]
• 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 (of the first and second kind):^[7][8]

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

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 ${\displaystyle n}$).^[9]

Fermat prime[edit]

Seventeen is the third Fermat prime, as it is of the form ${\displaystyle 2^{2^{n}}+1}$ with ${\displaystyle n=2}$.^[10] 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.^[11] It is one more than the smallest number with exactly seventeen divisors, 65,536 = 2¹⁶.^[12]

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.^[13][14]

Quadratic integer matrix[edit]

A positive definite quadratic integer matrix represents all primes when it contains at least the set of seventeen numbers:

${\displaystyle \{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,67,73\}.}$

Only four prime numbers less than the largest member are not part of the set (53, 59, 61, and 71).^[15]

Geometric properties[edit]

Two-dimensions[edit]

• There are seventeen crystallographic space groups in two dimensions.^[16] These are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper.

• Also in two dimensions, seventeen is the number of combinations of regular polygons that completely fill a plane vertex.^[17] Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42,^[18] 3.8.24,^[19] 3.9.18,^[20] 3.10.15,^[21] 4.5.20,^[22] and 5.5.10)^[23] exclusively surround a point in the plane and fill it only when irregular polygons are included.^[24]

• On the other hand, 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".^[25]

Seventeen is the minimum number of vertices on a two-dimensional graph such that, if the edges are colored with three different colors, there is bound to be a monochromatic triangle; see Ramsey's theorem.^[26]

Enumeration of icosahedron stellations[edit]

In three-dimensional space, there are seventeen distinct fully supported stellations generated by an icosahedron.^[27] The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules.^[28][29] 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).^[30] Seventeen distinct fully supported stellations are also produced by truncated cube and truncated octahedron.^[27]

Four-dimensional zonotopes[edit]

Seventeen is also the number of 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.^[31]

Abstract algebra[edit]

Seventeen is the highest dimension for paracompact Vineberg polytopes with rank ${\displaystyle n+2}$ mirror facets, with the lowest belonging to the third.^[32]

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.^[33] 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.^[34]

Other notable properties[edit]

• The sequence of residues (mod n) of a googol and googolplex, for ${\displaystyle n=1,2,3,...}$, agree up until ${\displaystyle n=17}$.
• Seventeen is the longest sequence for which a solution exists in the irregularity of distributions problem.^[35]

Complex analysis[edit]

There are seventeen 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.

Sudoku puzzle[edit]

The minimum possible number of givens for a sudoku puzzle with a unique solution is 17.^[36][37]

Spiral of Theodorus[edit]

17 is the least ${\displaystyle k}$ for the Theodorus Spiral to complete one revolution.^[38] This, in the sense of Plato, who questioned why Theodorus (his tutor) stopped at ${\displaystyle {\sqrt {17}}}$ when illustrating adjacent right triangles whose bases are units and heights are successive square roots, starting with ${\displaystyle 1}$. In part due to Theodorus’s work as outlined in Plato’s Theaetetus, it is believed that Theodorus had proved all the square roots of non-square integers from 3 to 17 are irrational by means of this spiral.

In science[edit]

• The atomic numberofchlorine.
• 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.^[39]

In languages[edit]

Grammar[edit]

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[edit]

• 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,^[40] and some jurisdictions around the world, 17 is the age of sexual consent^[41]
• 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.
• InTajikistan, North Korea and Timor-Leste, the age of majority

In culture[edit]

Music[edit]

Bands[edit]

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

Albums[edit]

• 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
• Sector 17, a 2022 repackaged album by Seventeen

Songs[edit]

• "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[edit]

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

Film[edit]

• Seventeen (1916), an adaptation of the novel of the same namebyBooth 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[edit]

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

Games[edit]

• 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[edit]

• 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.
• InThe 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[edit]

• 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.^[42]
• In the YasnaofZoroastrianism, 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 sports[edit]

• 17 is the number of the longest winning streak in NHL history, which the Pittsburgh Penguins achieved in 1993.
• Larry Ellison's victorious 2013 Americas Cup Oracle racing yacht bears the name "17".
• 17 is the number of individual laws mentioned in the Laws of the Game (association football).
• 17 is the number of games played by each NFL team as of 2021.
• Since the start of the 2014 season, Formula One drivers have been able to choose their own car number; however, following the fatal accident of Jules Bianchi, who drove car #17, the number was retired.

In other fields[edit]

Seventeen is:

• Described at MIT as 'the least random number', according to the Jargon File.^[43] This is supposedly because in a study where respondents were asked to choose a random number from 1 to 20, 17 was the most common choice.
  • This study has been repeated a number of times.^[44]
• The number of guns in a 17-gun salute to U.S. Army, Air Force and Marine Corps Generals, and Navy and Coast Guard admirals.
• The maximum number of strokes of a Chinese radical.
• The total number of syllables in a haiku (5 + 7 + 5).
• In the Nordic countries the seventeenth day of the year is considered the heart and/or the back of winter.
• "Highway 17" or "Route 17": See List of highways numbered 17 and List of public transport routes numbered 17.
• Seventeen, also known as Lock Seventeen, an unincorporated place in Clay Township, Tuscarawas County, Ohio.
• Seventeen was the former name of a yacht prior to being commissioned in the US Navy as the USS Carnelian (PY-19).

• InItalian 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 speciesofcicadas 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[edit]

• Berlekamp, E. R.; Graham, R. L. (1970). "Irregularities in the distributions of finite sequences". Journal of Number Theory. 2 (2): 152–161. Bibcode:1970JNT.....2..152B. doi:10.1016/0022-314X(70)90015-6. MR 0269605.

External links[edit]

Wikimedia Commons has media related to 17 (number).

Look up seventeen in Wiktionary, the free dictionary.

• Properties of 17
• Mathematical properties of 17 Archived 2019-07-29 at the Wayback Machine at yellowpigs.net
• 17
• is 17 the most random number at the wayback machine.
• Number 17 at the Database of Number Correlations
• Prime Curios for the number 17