In mathematics
17 is the seventh prime number, which makes seventeen the fourth super-prime, as seven is itself prime. The next prime is 19, with which it forms a twin prime.[1] It is a cousin prime with 13 and a sexy prime with 11 and 23.[2][3] It is an emirp, and more specifically a permutable prime with 71, both of which are also supersingular primes.[4][5]
Seventeen is the only prime number which is the sum of four consecutive primes: 2,3,5,7. Any other four consecutive primes summed would always produce an even number, thereby divisible by 2 and so not prime.
Seventeen can be written in the form and , and, as such, it is a Leyland prime and Leyland prime of the second kind:[6][7]
- .
17 is one of seven lucky numbers of Euler which produce primes of the form .[8]
Seventeen is the sixth Mersenne prime exponent, yielding 131,071.[9]
Seventeen is the third Fermat prime, as it is of the form , specifically with .[10] Since 17 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.[11][12]
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".[13]
Seventeen is the minimum number of vertices on a graph such that, if the edges are coloured with three different colours, there is bound to be a monochromatic triangle; see Ramsey's theorem.[14]
There are also:
- 17 crystallographic space groups in two dimensions.[15] 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.[16] Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42,[17] 3.8.24,[18] 3.9.18,[19] 3.10.15,[20] 4.5.20,[21] and 5.5.10)[22] exclusively surround a point in the plane and fill it only when irregular polygons are included.[23]
- 17 distinct fully supported stellations generated by an icosahedron.[24] The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules.[25][26] Without counting the icosahedron as a zeroth stellation, this total becomes 58, a count equal to the sum of the first seventeen prime numbers.[27]
- 17 distinct fully supported stellations are also produced by truncated cube and truncated octahedron.[24]
- 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.[28]
- 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.
Seventeen is the highest dimension for paracompact Vinberg polytopes with rank mirror facets, with the lowest belonging to the third.[29]
Seventeen is the minimum possible number of givens for a sudoku puzzle with a unique solution. This was long conjectured, and was proved between 2012 and 2014.[30][31]
The sequence of residues (mod n) of a googol and googolplex, for , agree up until .
A positive definite quadratic integer matrix represents all primes when it contains at least the set of 17 numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}.[32] Only four prime numbers up to 73 are not part of the set.
