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 and

Therefore but this is a contradiction. So two different pairs of natural numbers cannot have both the same sum and same product.