Thread: sums and products...

i suppose this is the correct place to post this question...it is easy to prove that 2 different pairs of Natural numbers cannot have the same sum AND the same product.Can this be proven for groups of 3,4,5,...n Natural numbers?

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.

Originally Posted by Lasombra

i suppose this is the correct place to post this question...it is easy to prove that 2 different pairs of Natural numbers cannot have the same sum AND the same product.Can this be proven for groups of 3,4,5,...n Natural numbers?

It doesn't work for sets of four numbers. The sets {1,4,4,6} and {2,2,3,8} each have sum 15 and product 96.