Number Puzzle

God picks two numbers a and b from {2, 3, ... , 100}, where a=b is possible, and gives Mr. S the sum S=a+b and Mr. P the product P=ab.  Now the following dialogue takes place: Mr. S says to Mr. P, "I do not know the numbers, but I know that you do not know."  Mr. P answers, "Then I know the numbers." Then Mr. S says, "Then I also know the numbers!"  What numbers has God chosen?

OK, so first off I'm going to presume that the trick lies in the prime numbers, so the prime numbers from 2-100 are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

The only way that Mr. P could know the numbers is if his number was composed of primes, so this suggests that Mr. S has a way of knowing that the numbers are not prime from his number.

One way Mr. S could know that the numbers are not both prime is that his number is odd and not 2 + another prime.  Meaning that his number is composed of at least one even number > 2 and one odd number.

When Mr. S tells Mr. P that he knows his number is not composed of primes, this lets Mr. P know that one number is even and the other is odd.  For Mr. P to know a & b from this fact requires something special about his number.  This suggests that one of the factors must be a prime number > 2.