2 problems

Let $\theta$ be the fixed point of $x\mapsto \xi^x$ and define the iterated power towers by $h_1(\xi)=\xi$ and $h_{k+1}(\xi)=\xi^{h_k(\xi)}$ for $k\geq 1$. An ultra-Liouville numbe…

Gelfond's Power Tower Conjecture. The number ${}^kz$ is transcendental for every $k\geqslant2$. Moreover, when $z:=e^{\omega}$, the numbers ${}^1z,{}^2z,{}^3z,\dots$ are algebraica…