Another characterization of the cofinality? Is it true that $cf(\kappa)=\min \{\lambda:\ \kappa^{\lambda}>\kappa\}$? 
$cf(\kappa)$ is certainly $\geq$ than that minimum since $\kappa^{cf(\kappa)}>\kappa$, but I don't know how to tackle the inverse inequality. What's worse, I'm not even sure if my hypothesis is true :) Is it?
(I'm actually fighting to prove that this $\min$ is a regular cardinal, and my idea was that in could be equal to the cofinality).

For infinite cardinal $k,$ it's also consistent with ZFC that it is true because of Konig's Theorem (aka Konig's Lemma): $k^{cf(k)}>k,$ and because GCH implies that $k^l=k$ for $0<l<cf(k).$
