Is $\ln(\ln(n))$ irrational for any integer $n>1$?

Is there $n\in \mathbb{N}$ such that $\ln(\ln(n)) \in \mathbb{Q}$? If such $n$ exists, we will get $$\ln(\ln(n)) = \frac{p}{q}, \quad p, q \in \mathbb{Z}.$$ Hence we will get $n = e^{e^{p/q}},$ where the question about the nature of $e^{e^{p/q}}$ haven't been answered yet.

Is there any other direction ? Thank you in advance.

Solution 1:

@GEdgar in a comment suggested that this is not known, and @Peter suggested that someone work out how it follows from

Schanuel's Conjecture: If $z_1,\dots,z_k$ are linearly independent over $\mathbb Q$, then $\mathbb Q(z_1,\dots,z_k,e^{z_1},\dots,e^{z_k})$ has transcendence degree at least $k$ over $\mathbb Q$.

So here we go:

Proposition: Schanuel's Conjecture implies that $\ln(\ln(n))\not\in\mathbb Q$ for all $n\in\mathbb N$ with $n>1$.

Proof: Suppose $\ln(\ln(n))=r\in\mathbb Q$, so that $e^{e^r}=n$. Let $z_1=r$ and $z_2=e^r$. Suppose $$az_1+bz_2=0,\qquad a,b\in\mathbb Q.$$ Then either $b=0$ or $e^r=-az_1/b\in\mathbb Q$. But by Schanuel's Conjecture with $k=1$, we have that $e^r$ is transcendental. So it must be that $b=0$. If $z_1=0$ then $e^{e^0}=n$ which is false (simply since $2<e<3$). We conclude that $(a,b)=(0,0)$ and so we have shown that $z_1, z_2$ are linearly independent over $\mathbb Q$.

So by Schanuel's Conjecture, $$\mathbb Q(z_1,z_2,e^{z_1},e^{z_2})=\mathbb Q(r,e^r,e^r,e^{e^r})=\mathbb Q(e^r,e^{e^r})$$ has transcendence degree at least 2 over $\mathbb Q$. But this implies that both $e^r$ and $e^{e^r}$ are transcendental numbers. $\Box$