Seventeen is the 7th prime number. The next prime is nineteen, with which it forms a twin prime.
17 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 compass and unmarked ruler. This was proven by Carl Friedrich Gauss.
There are exactly 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.
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".
17 is the least random number, according to the Hackers' Jargon File.
17 is the minimum possible number of givens for a sudoku puzzle with a unique solution. This was long conjectured, and was proved in 2012.
There are 17 orthogonal curvilinear coordinate systems (to within a conformal symmetry) in which the 3-variable Laplace equation can be solved using the separation of variables technique.
17 is the sixth Mersenne prime exponent, yielding 131071.
17 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. The next such number is 65.
17 is the minimum number of vertices on a graph such that, if the edges are coloured with 3 different colours, there is bound to be a monochromatic triangle. (See Ramsey's Theorem.)