Q:

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

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!

A:

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 $abc$.

