I picked four integer numbers, none negative.
If I had told you their product, you would have known what the numbers were.
If I had told you instead the sum of their squares, you would also have known what the numbers were.
But if I had told you instead the sum of the numbers, you wouldn't have been able to tell what the numbers were.
Which were the numbers?
There can be no zeroes, for then you wouldn't be able to deduce anything if you were given the product.
If the product had two prime factors P and Q, there could be many answers: whatever, whatever, P, Q, or whatever, whatever, 1, PxQ.
Thus, the numbers are 1, 1, 1, P, and P is prime.
The numbers cannot be 1, 1, 1, 2, for that would allow deducing them from their sum.
But if the numbers are 1, 1, 1, 3,
their product is unique, the sum of their squares is also unique, but the sum also allows 1, 1, 2, 2.