I am trying to figure out this impossible problem by Martin Gardener and still havent found a suitable solution or link to it

> 
> Two mathematicians S and P are discussing two
>  unknown integers, both greater than 1. S knows only the sum of the
>  numbers, whereas P knows only their product.:
> 
> 
> S: "I see no way you can determine my sum." 
>  P (after a suitable delay): "That didn't help me. I still don't know your sum." S (after
>  another delay): "Now I know your product." What are the two numbers?
> 
> 
> 

I have seen the answers on the xkcd wiki link for this at <http://wiki.xkcd.com/irc/Talk:Puzzles#The_Sam_And_Polly_Problem> say 13 and 16 and it checks out mathematically but then there are more ambiguous responses and most of the attempts seem brute force rather than logic.

My question is two fold

1. How to solve this problem logically?
2. Are there more answers then 13,16 or this answer is the only one even as the upper bound k(upper bound on answer) shifts higher