Mathematics
Seventeen is the seventh prime number, which makes it the fourth super-prime, [3] as 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, [4] a cousin prime with 13, [5] and a sexy prime with both 11 and 23. [6] Furthermore,
- It is the sixth Mersenne prime exponent for numbers of the form , yielding 131071. [7]
- 17 is one of six lucky numbers of Euler, the positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. [8]
- 17 can be written in the form and ; and as such, it is a Leyland prime (of the first and 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]
Fermat prime
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]
Quadratic integer matrix
A positive definite quadratic integer matrix represents all primes when it contains at least the set of seventeen numbers:
Only four prime numbers less than the largest member are not part of the set (53, 59, 61, and 71). [17]
Geometric properties
Two-dimensions
- There are seventeen crystallographic space groups in two dimensions. [18] 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. [19] Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42, [20] 3.8.24, [21] 3.9.18, [22] 3.10.15, [23] 4.5.20, [24] and 5.5.10) [25] exclusively surround a point in the plane and fill it only when irregular polygons are included. [26]
- 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. [27]
- 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". [28]
17 is the least for the Theodorus Spiral to complete one revolution. [29] This, in the sense of Plato, who questioned why Theodorus (his tutor) stopped at when illustrating adjacent right triangles whose bases are units and heights are successive square roots, starting with . 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. [30] The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules. [31] [32] 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). [33] Seventeen distinct fully supported stellations are also produced by truncated cube and truncated octahedron. [30]
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. [34]
Abstract algebra
Seventeen is the highest dimension for paracompact Vineberg polytopes with rank mirror facets, with the lowest belonging to the third. [35]
17 is a supersingular prime, because it divides the order of the Monster group. [36] 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. [37]
Other notable properties
- The sequence of residues (mod n) of a googol and googolplex, for , agree up until .[ citation needed ]
- Seventeen is the longest sequence for which a solution exists in the irregularity of distributions problem. [38]
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. [39] [40]
