There is this statement that GCH holds iff any pair of regular cardinals $\kappa,\lambda$ such that $\kappa<\lambda$ satisfy that $\lambda^\kappa = \lambda$.
Assume we do have two such cardinals. Take the sequence $\langle\lambda^\xi\mid \xi<\kappa\rangle$, then it seems that $\lim_{\xi\to\kappa} \lambda^\xi = \lambda^\kappa = \lambda$ and so $\operatorname{cf}(\lambda) \le \kappa < \lambda$ contradicting regularity.
I am obviously missing something, but I have no idea what, because if my reasoning is correct it seems that under these hypotheses there could not be two different regular cardinals.
Thanks in advance!
Your mistake is that under the working assumption, $\lambda^\xi=\lambda$. So the sequence is constant. Therefore the cofinality of $\lambda$ is not $\kappa$ or less, because that would require a sequence of cardinals smaller than $\lambda$ itself.