17 is a
Leyland number and
Leyland prime, using 2 and 3 (23 + 32), and using 3 and 4 (34 - 43). 17 is a
Fermat prime. 17 is one of six
lucky numbers of Euler. 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. The minimum possible number of givens for a
sudoku puzzle with a unique solution is 17.
Geometric properties Two-dimensions , with a maximum
right triangles laid edge-to-edge before one revolution is completed. The largest triangle has a
hypotenuse of \sqrt {17}. • 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 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.
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 a
supersingular prime, because it divides the order of the
Monster group. 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 ) 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. s in the
Standard Model of physics. == Other fields ==