If $g$ is a primitive root of $p$ (i.e. $\mathbb{F}_p^{\times}=\langle g \rangle$) show that two consecutive powers of $g$ have consecutive least residues. That is, show that there exists $k$ such that $g^{k+1} \equiv g^k + 1 \pmod{p}$. Can this be generalized to an arbitrary finite cyclic group?
Thank you so much.
Since $g-1$ is invertible in the field, then there is $x=g^k$ such that $x(g-1)=1$.