How can I make $a^{21}+b^{21}+c^{21}$ out of $a+b+c$, $a^2+b^2+c^2$, and $a^3+b^3+c^3$?

Question

How can I make $a^{21}+b^{21}+c^{21}$ out of $a+b+c$, $a^2+b^2+c^2$, and $a^3+b^3+c^3$? Basically, I'm wondering how to manipulate the three expressions to get exactly $a^{21}+b^{21}+c^{21}$. However, since the exponents are so large, it is difficult to see how to manipulate them. Any help? Thanks!

Answer 1

Let $a+b+c=x$, $a^2+b^2+c^2=y$ and $a^3+b^3+c^3=z$.

Firstly, we can get $ab+ac+bc=\frac{1}{2}(x^2-y)$ and $abc$ from $$z=x^3-3x\cdot\frac{1}{2}(x^2-y)+3abc.$$

Now, since $a^{21}+b^{21}+c^{21}$ is a symmetric polynomial,

we can write it like polynomial of $a+b+c$, $ab+ac+bc$ and $abc$.

Answer 2

Let $s(n)=a^n+b^n+c^n$. The sequence has a recursion based on the polynomial $$(x-a)(x-b)(x-c)=x^3-e_1x^2+e_2x-e_3$$ where $e_1,e_2,e_3$ are the elementary symmetric functions. The recursion is $$s(n)=e_1s(n-1)-e_2s(n-2)+e_3s(n-3).$$ Using this recursion you can find any $s(n)$ given $s(1),s(2),s(3)$.