During my research, I come across this question.
Let $p=3^{101}-4$, $x_0=0$ and $x_{n+1}=x_n^3+1 \mod p$.
Is it true that $\exists k \in \mathbb N \cap [3,p-2], x_k=1$ ?
I have posted this question here : https://artofproblemsolving.com/community/c6h3213569
Since $p$ is prime and $\left(\frac{-3}p\right)=-1$, the function $f(x) := x^3 +1$ has the unique inverse and thus represents a permutation on $\mathbb Z/p\mathbb Z$. This means that starting at $x_1=1$ iterative application of $f$ should yield $1$ again. The number of iterations equals the length of the cycle to which $1$ belongs in the permutation $f$.
It remains to notice that $f$ has one fixed point and one cycle of length 3 as established by this PARI/GP code. It follows that the cycle of element 1 has length $\leq p-4$ giving $k$ in the desired interval.
Btw, the inverse of $f$ can be stated explicitly as $f^{-1}(y)=(y-1)^{2\cdot 3^{100}-3}$.
