Mr. S. and Mr. P.

From rec.puzzles

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:

• Mr. P.: I do not know the two numbers.
• Mr. S.: I knew that you didn't know the two numbers.
• Mr. P.: Now I know the two numbers.
• Mr. S.: Now I know the two numbers.

Given that the above statements are absolutely truthful, what are the two numbers?

Solution