<s>[INST] Roots of a Complex Number expression If $z_1$, $z_2$, $z_3$, $z_4$ are roots of the equation $z^4+z^3+z^2+z+1=0$,
then what is the least value of $\lfloor mod(z_1 + z_2)\rfloor + 1$? 
($\lfloor.\rfloor$ denotes Greatest Integer Function)
 [/INST] $$z^4+z^3+z^2+z+1=\frac{z^5-1}{z-1}=0$$
$$\therefore z^5=1\text{ and } z\ne1$$
These solutions can be plotted on an Argand diagram which more easily shows that the solution is given by the two complex conjugates with the smallest magnitude real parts - $e^{\frac{2i\pi}5}$ and $e^{\frac{-2i\pi}5}$. Adding these values gives
$$e^{\frac{2i\pi}5}+e^{-\frac{2i\pi}5}=\cos{\left({\frac{2\pi}5}\right)}+i\sin{\left({\frac{2\pi}5}\right)}+\cos{\left({\frac{2\pi}5}\right)}-i\sin{\left({\frac{2\pi}5}\right)}=2\cos{\left({\frac{2\pi}5}\right)}$$
So the smallest value is given by
$$\lfloor z_1+z_2\rfloor+1=\left\lfloor 2\cos{\left({\frac{2\pi}5}\right)}\right\rfloor+1=\lfloor 0.618...\rfloor+1=1$$
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