plz find entier numbers n and p which make the following system to have solutions
z^n=1
(1+z)^p=1
Those equations imply that both z and 1+z lie on the unit circle. The only such points are $\displaystyle z = -\tfrac12\pm\tfrac{\sqrt3}2i$, so that n=3 and p=6 (or multiples thereof).
Under the hypothesis that $\displaystyle 0 \in \mathbb {Z}$, $\displaystyle z=0$, $\displaystyle n=0$ and $\displaystyle p \in \mathbb{Z}$ arbitrary is also solution...