In mathematics
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 ,[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 and ; and as such, it is a Leyland prime and Leyland prime of the second kind:[9][10]
The number of integer partitions of 17 into prime parts is 17 (the only number such that its number of such partitions is ).[11]
Seventeen is the third Fermat prime, as it is of the form with .[12] 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.[13] It is one more than the smallest number with exactly seventeen divisors, 65,536 = 216.[14] 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.[15][16]
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]
Seventeen 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.[18]
There are also:
- Seventeen crystallographic space groups in two dimensions.[19] These are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper.
- Seventeen combinations of regular polygons that completely fill a plane vertex.[20] Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42,[21] 3.8.24,[22] 3.9.18,[23] 3.10.15,[24] 4.5.20,[25] and 5.5.10)[26] exclusively surround a point in the plane and fill it only when irregular polygons are included.[27]
- 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.
- Seventeen distinct fully supported stellations generated by an icosahedron.[28] The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules.[29][30] 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).[31]
- Seventeen distinct fully supported stellations are also produced by truncated cube and truncated octahedron.[28]
- Seventeen 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.[32]
Seventeen is the highest dimension for paracompact Vineberg polytopes with rank mirror facets, with the lowest belonging to the third.[33]
Seventeen is the longest sequence for which a solution exists in the irregularity of distributions problem,[34] while the sequence of residues (mod n) of a googol and googolplex, for , agree up until .
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.[35] 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.[36]
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).[37]
