We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
We consider the application $f_\alpha$ defined on $\mathbb{R}_+^*$ by $f_\alpha(x) = x^\alpha$. Show that $\alpha$ is a natural number if and only if one of the successive derivatives of $f_\alpha$ vanishes at least at one strictly positive real.