Are there any $k\gt2$ for which we have a solution of the nonlinear diophantine equation $x^{k-1}=\sum_{i=0}^{k-1}10^i$.
This question arose when I tried to provide a simple solution to this question Find matrix $A\in \mathcal{M}_n (\mathbb{N})$ such that $A^k =\left( \sum_{i=1}^{k}10^{i-1} \right)A$. for at least some $k\in[2,\infty)\cap\Bbb N$.
I tried the first nine or ten, and there were none.
Taking @Jaap Scherphuis' suggestion, note that $10^{k-1}\lt\underbrace{111\dots1}_{\text{k-times}}\lt11^{k-1}$.
The first inequality is obvious. For the second, just apply the binomial theorem to $(10+1)^{k-1}$.
Thus there are no solutions.