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. The next prime is nineteen, with which it forms a twin prime.
Seventeen is a permutable prime, a Leyland prime of the second kind and a supersingular prime.
Seventeen is the third Fermat prime, as it is of the form 22n + 1, specifically with n = 2. 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.
There are 17 two-dimensional space (plane symmetry) groups. These are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper.
There are 17 two-dimensional combinations of regular polygons that completely fill a plane vertex. 11 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.
There are 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.
Seventeen is the highest dimension for paracompact Vinberg polytopes of rank n+2, with the lowest belonging to the third.
Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, for all positive n < p − 1.
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 possible number of givens for a sudoku puzzle with a unique solution. This was long conjectured, and was proved in 2012–14.
There are 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 sixth Mersenne prime exponent, yielding 131071.
Seventeen is the first number that can be written as the sum of a positive cube and a positive square in two different ways; that is, the smallest n such that x3 + y2 = n has two different solutions for x and y positive integers ((1,4) or (2,3)). The next such number is 65.
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.)
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.
The sequence of residues (mod n) of a googol and googolplex, for n = 1, 2, 3, ..., agree up until n = 17.
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.
In Italian, 17 is also the first compound number (diciassette), whereas sixteen is sedici.
Main article: 17 (disambiguation) § Music