Seventeen is the 7th prime number. The next prime is nineteen, with which it forms a twin prime. 17 is the sum of the first four primes. 17 is the sixth Mersenne prime exponent, yielding 131071. 17 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1.

17 is the third Fermat prime, as it is of the form 24 + 1, and it is also a Proth prime. Since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss.  Another consequence of 17 being a Fermat prime is that it is not a Higgs prime for squares or cubes; in fact, it is the smallest prime not to be a Higgs prime for squares, and the smallest not to be a Higgs prime for cubes.

17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime.

As 17 is the least prime factor of the first twelve terms of the Euclid–Mullin sequence, it is the thirteenth term.

Seventeen is the aliquot sum of two numbers, the odd discrete semiprimes 39 and 55 is the base of the 17-aliquot tree.

There are exactly seventeen 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.