# seventeen

Seventeen is the number most often picked in response to the request "Pick a random number
from 1 to 20." Seventeen is a **Fermat prime** (a prime
number of the form 2^{2n} + 1, where *n* is a positive
integer), the exponent of a **Mersenne prime** (a prime *p* for which 2* ^{p}* - 1 is prime), and the only prime that is
the sum of four consecutive primes (2 + 3 + 5 + 7). Seventeen is also the
smallest number for which the sum of the digits of its cube is equal to
the number: 173 = 4913, 4 + 9 + 1 + 3 = 17, and the smallest number that
can be written as A

^{2}+ B

^{3}in two different ways: 17 = 3

^{2}+ 2

^{3}= 4

^{2}+ 1

^{3}. The pair (8, 9), whose sum is 17, is the only pair of consecutive numbers where one is a square and the other is a cube (a result proved by Leonhard Euler.) There are 17 planar crystallographic groups. The minimum number of faces on a convex polyhedron that has only one stable face is 17. (A stable face is one that the figure can rest on without falling over; most polygons have more than one such face.) Seventeen is also the answer to the follow problem: At a party where any two people have previously met each other in one of three other places, what is the least number of people who must be at the party to guarantee that there is at least one group of three people who have met each other before in the same place?