# Is knowing the Sum and Product of k different natural numbers enough to find them?

Can we uniquely identify the set of k different natural numbers (no two are the same) by knowing only their sum and product (and the value of k itself)?

• I suggest you try examples, with k equal to 3. – Mariano Suárez-Álvarez Mar 25 '16 at 2:02
• @MarianoSuárez-Alvarez: I don't see an obvious 3-number counter-example if $0$ is not a natural number. – user21820 Mar 25 '16 at 3:14
• @user21820 Agreed. Though there are 3-number counterexamples, they take a while to find by hand. – Bamboo Mar 25 '16 at 4:09
• It's possible for $k=2$. Not for $k \ge 3$. – TheRandomGuy Mar 25 '16 at 4:58
• Think about polynomials and their coefficients... This would imply that you could find all the roots of a polynomial by simply looking at the first and last coefficient. – Aleks J Mar 25 '16 at 11:14

• First (I think) triple example is: $$6\times8\times25=1200\text{ and }6+8+25=39$$ $$5\times10\times24=1200\text{ and }5+10+24=39$$ $$4\times15\times20=1200\text{ and }4+15+20=39$$ – Ian Miller Mar 26 '16 at 15:50
I think the minimal counterexample (in the sense of minimal sum) are the triples $\{2,8,9\}$ and $\{ 3,4,12 \}$ for which we have $$2+8+9 = 19 \qquad 2\cdot 8\cdot 9 = 144$$ and $$3 + 4 + 12 = 19 \qquad 3 \cdot 4 \cdot 12 = 144.$$