Quote Originally Posted by zaza
But the point is that S knows that P doesn't know. He is certain of the fact. He couldn't be certain if there was the possibility that the number in question could be made up of two primes.

In the instance below, if S had 28 then P could have 23 x 5 = 115, and if P had been given 115 then he would know what m+n was. Hence if S had 28, he couldn't be sure that P didn't already know what the two numbers were.
Exactly and I have checked the solution though have not proved it's unique as that will be time consuming without a computer