Show that there exists a set A of positive integers with the following property: for any infinite set S of primes, there exist two positive integers m in A and n not in A, each of which is a product of k distinct elements of S for some k ≥ 2.
Solution
Let the primes be p1 < p2 < p3 < ... . Let A consists of all products of n distinct primes such that the smallest is greater than pn. For example: all primes except 2 are in A; 21 is not in A because it is a product of two distinct primes and the smallest is greater than 3. Now let S = {pi1, pi2, ... } be any infinite set of primes. Assume that pi1 < pi2 < ... . Let n = i1. Then pi1pi2 ... pin is not in A because it is a product of n distinct primes, but the smallest is not greater than pn. But pi2pi3 ... pin+1 is in A, because it is a product of n distinct primes and the smallest is greater than pn. But both numbers are products of n distinct elements of S.
© John Scholes
jscholes@kalva.demon.co.uk
30 Oct 1998
Last corrected/updated 25 Aug 03