Prove that $1280000401$ is Composite I tried to prove $N=1280000401$ as composite using complex cube roots of unity:
we can write $$N=1+400+(128*10^{7})$$ which gives
$$N=1+20^2+20^{7}$$
now if $F(x)=1+x^2+x^7$, $w$ and $w^2$ are roots of $F(x)=0$ where $w=\frac{-1+i\sqrt{3}}{2}$ and $w^2$ its conjugate.
Hence $x^2+x+1$ is factor of $1+x^2+x^7$
Hence $1+20+20^2=421$ is a factor of $N$ and hence it is composite.
But how can we prove that without using complex numbers?

You can say $1+x^2+x^7=1+x+x^2+(x^7-x)=1+x+x^2+x(x^3-1)(x^3+1)=(1+x+x^2)(1+x(x-1)(x^3+1))$
but your approach is a good one to find this.  People have been very clever in finding things to try.
