For 2 different pairs of natural numbers we can do it without much difficulty. Let our pairs be and and remember these sets are not equal.
Suppose that and . Since we can say for some non-zero number
To ensure we then know that
Then and since we know that we must have
So by factoring we have . Remember that and so which means
Therefore but this is a contradiction. So two different pairs of natural numbers cannot have both the same sum and same product.