Can one find a set of n distinct positive integers such that the geometric mean of any (non-empty, finite) subset is an integer? Can one find an infinite set with this property?
Solution
Answer: yes, no.
Take each member to be an n! power (for example, 1n!, 2n!, ... , nn!).
Suppose we could find an infinite set. Take any two members m and n. Then for sufficiently large k, (m/n)1/k must be irrational. But now if we take any other a1, a2, ... , ak-1 in the set, (m a1 ... ak)1/k and (n a1 ... ak)1/k cannot both be integers. Contradiction.
© John Scholes
jscholes@kalva.demon.co.uk
25 Aug 2002