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.

17 was described at MIT as "the least random number", according to the Jargon File.

Mathematics
Seventeen is the seventh prime number, which makes it the fourth super-prime, as 7|seven is itself prime.

Prime properties
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, a cousin prime with 13, and a sexy prime with both 11 and 23. Furthermore,


 * It is the sixth Mersenne prime exponent for numbers of the form $$2^{n} - 1$$, yielding 131071.
 * It is also one of six lucky numbers of Euler $$n$$ which produce primes of the form $$m^{2}-m+n$$ for $$m=0, \ldots, n-1.$$ (I.e. for $$n$$ of 17 and $$m$$ of 16 there is 257.)
 * 17 can be written in the form $$x^y + y^x$$ and $$x^y - y^x$$; and as such, it is a Leyland prime (of the first and second kind):
 * $$2^{3} + 3^{2} = 17 = 3^{4} - 4^{3}.$$

The number of integer partitions of 17 into prime parts is 17 (the only number $$n$$ such that its number of such partitions is $$n$$).

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

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.

Quadratic integer matrix
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).

Two-dimensions



 * There are seventeen 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.


 * Also in two dimensions, seventeen is the number of 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.


 * 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.


 * 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".

17 is the least $$k$$ for the Theodorus Spiral to complete one revolution. This, in the sense of Plato, who questioned why Theodorus (his tutor) stopped at $$\sqrt{17}$$ when illustrating adjacent right triangles whose bases are 1|units and heights are successive square roots, starting with $$1$$. 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.

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

Four-dimensional zonotopes
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.

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

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. 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.

Other notable properties

 * The sequence of residues (mod $n$) of a googol and googolplex, for $$n=1, 2, 3, ...$$, agree up until $$n=17$$.
 * Seventeen is the longest sequence for which a solution exists in the irregularity of distributions problem.

Complex analysis
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
The minimum possible number of givens for a sudoku puzzle with a unique solution is 17.

Physics
Seventeen is the number of elementary particles with unique names in the Standard Model of physics. This is the same as the number of elementary particles that are their own antiparticle (i.e. from four of five classes of scalar or vector bosons, except for the W bosons) and those that are not (collectively, twelve quarks and leptons, and W); that is, without distinguishing between eight gluons, or two W± bosons as each other's antiparticles, W+ and W−. Gluons have color charges as force carriers, where they carry color charge and anti color charge, and gluon-gluon "annihilation" with opposing color charges can at times occur and produce pairs of photons (though not directly), for example. However, this interaction is not in the same vein of an "antiparticle", as defined in the SM.

Chemistry
Group 17 of the periodic table is called the halogens. The atomic number of chlorine is 17.

Biology
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).

In religion

 * In the Yasna of Zoroastrianism, seventeen chapters were written by Zoroaster himself. These are the five Gathas.
 * The number of surat al-Isra in the Qur'an is seventeen, at times included as one of seven Al-Musabbihat. 17 is the total number of Rak'as that Muslims perform during Salat on a daily basis.

Other fields
Seventeen is:
 * The total number of syllables in a haiku (5 + 7 + 5).
 * The maximum number of strokes of a Chinese radical.

Music
Where Pythagoreans saw 17 in between 16 from its Epogdoon of 18 in distaste, the ratio 18:17 was a popular approximation for the equal tempered semitone (12-tone) during the Renaissance.