Solution: The X and Y Dialogue

A: I don’t know the values x and y.
This means that X and Y cannot both be primes. 

B: I knew you didn’t know. I don’t know them either.
The first sentence is of primary value here. It indicates that the sum x+y that has been given to B cannot be split as the sum of two primes. This immediately says that x+y cannot be even since Goldbach’s conjecture (every even number greater than 2 can be written as the sum of two primes) has been verified for all numbers up to 400,000,000,000,000,000. So the sum x+y is odd and is not 2 greater than a prime. The full list of values x+y in the given range that satisfy these criteria is 11, 17, 23, 27, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87, 89, 93, 95, 97, 101, 107, 113, 117, 119, 121, 123, 125, 127, 131, 135, 137, 143, 145, 147, 149, 155, 157, 161, 163, 167, 171, 173, 177, 179, 185, 187, 189, 191, and 197.

A: Now I know x and y.
This statement tells us that among all of the proper factorizations of the product xy, only one of the factor pairs has a sum that is on the above list.

B: Now I know them too.
This statement tells us that the sum x+y can only be split in one way that satisfies the previous criterion.

For example, if the sum x+y were 11, then there would be four cases to consider: (x,y)=(2,9), (x,y)=(3,8), (x,y)=(4,7), and (x,y)=(5,6). If (x,y)=(2,9), then the product given to A would be 18 which satisfies the prior criterion since (x,y)=(2,9) is the only proper factorization of xy=18 that yields a sum, x+y=11, that is on the above list of possible sums. If (x,y)=(3,8), then the product 24 would also satisfy the above criterion since (x,y)=(3,8) is its only proper factorization of 24 that would produce a sum, 11, that is on the above list. Similarly, (x,y)=(4,7) satisfies the above criterion although (x,y)=(5,6) does not (since 30 can be written as the product of 2 and 15 and also as the product of 5 and 6, and their respective sums, 17 and 11, are both on the above list). Thus, three of the four cases satisfy the prior criterion, and we are seeking a value for which only a single split satisfies the above criterion.

Next consider the case where x+y=17. There are seven splits to examine. Of these seven, six fail to meet the above criterion (2×15 can also be written as 5×6, 3×14 can also be written as 2×21, 5×12 can also be written as 3×20, 6×11 can also be written as 2×33, 7×10 can also be written as 2×35, and 8×9 can also be written as 3×24). The only surviving pair is (x,y)=(4,13). Thus, x+y=17 is consistent with the entire dialogue with (x,y)=(4,13) as its satisfying pair.

Exhaustive analysis of the remaining possible sums shows that 17 is the only sum that has only one satisfactory split.

The answer is therefore x=4 and y=13.