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This occurred to me while looking at my answer to Looking for a Simple Proof of the Divergence of the Prime Harmonic Series

Show that $\ln(\ln(n))$ is not rational (and is undoubtedly transcendental) for all integers $n \ge 2$.

I am sure that this is true and have no idea how to prove it.

If it were, then $n = e^{e^{a/b}}$, and I don't see how to manipulate this to get a contradiction.

Gelfand-Schneider doesn't seem to apply.

There may be a theorem that $e^{e^r}$ is transcendental for rational $r$, but I don't know it.

If you can do this, then do the $k$-fold iterated log, $\ln_k$, to show that $\ln_k(n)$ is irrational (and, again, undoubtedly transcendental) for all $n$ large enough so that $\ln_k(n)$ is real.

Blue
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marty cohen
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    Have you checked out: https://math.stackexchange.com/questions/2063576/is-ln-lnn-irrational-for-any-integer-n1 ? It may or may not be of help. – DDS Jul 24 '19 at 17:33
  • Thanks. Essentially the same question. Not surprising.

    The answer there shows that it follows from https://en.wikipedia.org/wiki/Schanuel%27s_conjecture, which, of course, is not proven.

     – marty cohen Jul 24 '19 at 17:45
  • One of the reasons why the link `may or may not be of help.' – DDS Jul 24 '19 at 17:47

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