logical deduction - Product and Sum

My fellow mathematicians Paul and Sam are very fond of mathematical puzzles. One day, I came up with the idea on mathematical puzzles while three of us having lunch together.

I: Hey, guys. I am thinking of two integers which are greater than or equal to 2. I will tell you the product and the sum of two numbers. Why don't you guess what two numbers are?

P: If you tell us the product and the sum, finding two integers is a piece of cake.

I: I will tell Paul only the product and Sam only the sum.

I whispered the product to Paul and the sum to Sam.

I: First, I will ask Paul. Could you figure out two numbers?

P: (after thinking) I can't. I don't know what two integers are.

I: (to Sam) How about you Sam? Do you know what two unknown numbers are?

S: I do not know either.

Surprisingly, right after Sam answered that he does not know, Paul said;

P: Now, I think I know these two numbers.

S: I also found out two numbers.

Now, a question to the reader. What are two integers?

Remark: The answer is NOT 4 and 13.

From this riddle, after answering "I don't know" twice, the answer " I know" came one after another.

If "I don't know" comes three times and then answers "I know", what would two integers be?

What is the largest possible number of "I don't know" before one can answer "I know" when solving two integers?

Answer

The numbers are

6 and 2

Explanation

P gets 12 and S gets 8
P knows the numbers are (4, 3) or (6, 2), so replies I don't know
S knows the numbers are (2, 6) or (4, 4) but can not be (3, 5) as P doesn't know the answer, so still replies I don't know
Suddenly P realizes that numbers can not be (4, 3) otherwise S would have ruled out (5, 2) and guessed (4, 3) and hence (6, 2) is the answer so he replies I know the answer
S also realizes that the numbers can not be (4, 4) otherwise P could not know the number so he concludes the numbers are (6, 2) and replies I know the answer

I assume that both P and S are perfect logicians.

Edit:

Answers to the bonus questions are:

(4,4) for sequence no, no, no, yes
There may be infinite rounds of NO

I will add the details once I put down my paper work in text.