Let $A$ and $B$ be two invertible complex matrices. Suppose there exists an integer $N$ such that
$$A^k\sim B^k\quad\forall k\ge N$$
I would like to prove that $A$ is similar to $B$. I've tried to approaching this using Jordan Canonical Forms, and here is what I have done so far:
Let the distinct eigenvalues of $A$ be $\lambda_1,\ldots,\lambda_s\in\mathbb{C}^\times$ since $A$ is invertible. The Jordan Canonical Form (JCF) of $A$ can be written as $\text{diag}\{J_{n_1}(\lambda_1),\ldots,J_{n_s}(\lambda_s)\}$, where $J_{n_i}(\lambda_i)=\text{diag}\{J(k_{i1},\lambda_i),\ldots,J(k_{it_i},\lambda_i)\}$ and
$$J(k_{ij},\lambda_i)=\begin{pmatrix}\lambda_i\\1&\lambda_i\\&\ddots&\ddots\\&&1&\lambda_i \end{pmatrix}_{k_{ij}\times k_{ij}}$$
and thus the JCF of $A^k$ becomes $\text{diag}\{J_{n_1}(\lambda_1^k),\ldots,J_{n_s}(\lambda_s^k)\}$, which is a standard conclusion for invertible matrices. Since $A^N\sim B^N$ and $B$ is invertible, we can set $B$ is similar to $\text{diag}\{J_{n_1}(\mu_1),\ldots,J_{n_s}(\mu_s)\}$ because of the above conclusion, which implies the sizes and numbers of $B^k$'s Jordan blocks are the same as $k$ varies.
At this point, I need to prove that $\lambda_i=\mu_i$ for all $i\in\{1,\ldots,s\}$. Since that $A^N$ is similar to $B^N$, we can set $J_{n_i}(\lambda_i^N)=J_{n_i}(\mu_i^N)\Leftrightarrow\lambda_i^N=\mu_i^N$ for each $i\in\{1,\ldots,s\}$ without loss of generality. However, due to the presence of the unit roots, it seems we cannot directly conclude that $\lambda_i=\mu_i$; instead, it's possible that $\lambda_i^{N+1}=\mu_j^{N+1}$ for some $j\not=i$. But according to the pigeonhole principle, we can get that
$$\forall i\in\{1,\ldots,s\}\exists j\in\{1,\ldots,s\}:\forall M\exists k\ge M\,s.t.\,\lambda_i^k=\mu_i^k$$
since there must have at least one index $j$ been chosen infinitely times by given index $i$. But the proof is still incomplete.
I came across a solution that claims there are infinitely many prime numbers $p$ for which the above holds, and this fact can be used to eliminate the influence of the unit roots, thereby reducing $\lambda_i=\mu_i$ eventually. However I don't fully understand how to use it. I don't even know how to ensure the existentiality.
So my main question is whether this proposition is true, and if so, how to complete the proof starting from my current work (any other rigorous proof is welcome). Additionally, I would appreciate more details on the solution I mentioned above.
Thank you for any insights!
