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# 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 22n + 1, where n is a positive integer), the exponent of a Mersenne prime (a prime p for which 2p - 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 A2 + B3 in two different ways: 17 = 32 + 23 = 42 + 13. 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?