Usenet Newsgroups : rec.puzzles Hall of FameThe rec.puzzles Hall of Fame is a compilation of over 500 of the most popular puzzles that have been posted and discussed in the rec.puzzles newsgroup. In most cases a detailed solution has been provided. Many of these puzzles also appear in Braingle's own collection. 

Mr. S. and Mr. P. are both perfect logicians, being able to correctly deduce any truth from any set of axioms. Two integers (not necessarily unique) are somehow chosen such that each is within some specified range. Mr. S. is given the sum of these two integers; Mr. P. is given the product of these two integers. After receiving these numbers, the two logicians do not have any communication at all except the following dialogue: <<1>> Mr. P.: I do not know the two numbers. <<2>> Mr. S.: I knew that you didn't know the two numbers. <<3>> Mr. P.: Now I know the two numbers. <<4>> Mr. S.: Now I know the two numbers. Given that the above statements are true, what are the two numbers?
Solution
Follow Braingle!