17 (number)

Seventeen is the seventh prime number, which makes it the fourth super-prime,[1] as seven is itself prime. It forms a twin prime with 19,[2] a cousin prime with 13,[3] and a sexy prime with both 11 and 23.[4] 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 is one of six lucky numbers of Euler which produce primes of the form ${\displaystyle k^{2}-k+41}$,[5] and the sixth Mersenne prime exponent, which yields 131,071.[6] It is also the minimum possible number of givens for a sudoku puzzle with a unique solution.[7][8] 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 and Leyland prime of the second kind:[9][10]

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

17 is the third Fermat prime, as it is of the form ${\displaystyle 2^{2^{n}}+1}$ with ${\displaystyle n=2}$.[11] 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.[12] It is one more than the smallest number with exactly seventeen divisors, 65,536 = 2¹⁶.[13] 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.[14][15]

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".[16]

17 is the minimum number of vertices on a graph such that, if the edges are colored with three different colors, there is bound to be a monochromatic triangle; see Ramsey's theorem.[17]

There are also:

• 17 crystallographic space groups in two dimensions.[18] 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.[19] Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42,[20] 3.8.24,[21] 3.9.18,[22] 3.10.15,[23] 4.5.20,[24] and 5.5.10)[25] exclusively surround a point in the plane and fill it only when irregular polygons are included.[26]

• 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.[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]

17 distinct fully supported stellations are also produced by truncated cube and truncated octahedron.[27]

• 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.[31]

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

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

In abstract algebra, 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]

A positive definite quadratic integer matrix represents all primes when it contains at least the set of seventeen numbers: {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).[35]

• 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.[36]

• 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,[37] and some jurisdictions around the world, 17 is the age of sexual consent[38]
• 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

• 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 the record for most NBA championships in NBA History, which the Boston Celtics (and as of 2020, the Los Angeles Lakers) achieved.
• 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.

Seventeen is:

• Described at MIT as 'the least random number', according to the Jargon File.[40] 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.[41]
• 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).

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 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 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 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.

• 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.

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