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


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.


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