Семнадцать: число и его свойства

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.

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

Seventeen is the third Fermat prime, as it is of the form with 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.

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. 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". 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.

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.

Seventeen is the highest dimension for paracompact Vineberg polytopes with rank 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.

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

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.

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

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 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.

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 of Zoroastrianism, 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.

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.