Find three numbers given their sum, product and sum of their squares

Question

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z

Answer 1

Here is a method you can try. Let

$s_1=a+b+c$,

$s_2=a^2+b^2+c^2$,

$p_2=ab+bc+ac$,

$p_3=abc$

$a, b, c$ are the roots of the cubic $$0=(x-a)(x-b)(x-c)=x^3-s_1x^2+p_2x-p_3$$

We don't know $p_2$ but can calculate it using $s_1^2=s_2+2p_2$, and then solve the cubic to find $a,b,c$.

Answer 2

Yes. Try squaring $X=(a+b+c)$.