Let's start by giving some definitions that are very important.
We call a number NonPrimeSum
(NPS), if it cannot be the sum of two prime numbers.
(ex. 11, 17, 23, 27, 29, 35, 37 ...) (11 = 9+2, 8+3, 7+4,
6+5)
We call a number Alpha, if there are
ONLY two numbers whose product is equal
to this number and whose sum is an NPS number.
(ex. 18, 52 ...) (18 = 2*9, 3*6 and 2+9=11
an NPS number and 3+6=9 not an NPS
number)
We call an NPS number Epsilon,
if there are ONLY two mumbers whose sum is equal
to this number and whose product is an Alpha
number.
So from the conversation we can derive that:
In order for A to say to B that he doesn't know the numbers, the
numbers cannot be prime
numbers, so their sum must be an NPS
number.
In order for B to say to A that he knows the numbers, their product must be an Alpha number.
And in order for A to say to B that he knows the numbers too, their
sum must be an Epsilon
number.
17 = 2+15, 3+14, 4+13, 5+12, 6+11, 7+10, 8+9
2*15 = 30, (2*15, 3*10, 5*6) with sums (17,
13, 11). So 30 is not an Alpha
number.
3*14 = 42, (2*21, 3*14, 6*7) with sums (23,
17,
13). Also not an Alpha number.
4*13 = 52, (2*26,
4*13) with sums (28,
17).
So 52 is an Alpha number.
5*12 = 60, (2*30, 3*20, 4*15, 5*12, 6*10) with
sums (32, 23, 19, 17,
16).
6*11 = 66, (2*33, 3*22, 6*11)
with sums (35, 25, 17).
7*10 = 70, (2*35, 5*14, 7*10)
with sums (37, 19, 17).
8* 9 = 72, (2*36, 3*24, 4*18, 6*12, 8*9)
with sums (38, 27, 22, 18, 17).
So 17 is an Epsilon number because there
are only two numbers (4, 13) whose sum is 17
and whose product 52 is an Alpha number.
So the numbers are .
(This is one of many solutions)
11 = 2+9, 3+8 ...
2*9 = 18, (2*9, 3*6)
with sums (11, 9).
18 is an Alpha number.
3*8 = 24, (2*12, 3*8, 4*6)
with sums (14, 11, 10). 24 is
also an Alpha number.
So there are more than two numbers whose sum is 11 and whose product
is an Alpha number.
And by definition, 11 is not an Epsilon
number.