In this question, I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this?

After several years, it appears this is an open problem. As a non-number theorist, I had assumed there would be known results that would answer the question. I was aware of the difficulty in proving various constants to be transcendental -- such as $e + \pi$, which is not known to be transcendental at present.

However, I was looking at a question that seems simpler, naively: whether a number is an integer, rather than whether it is transcendental. It seems that what appeared to be possibly simpler is actually not, with current techniques.

The main motivation for asking about this particular number is that it is very large. It is certainly possible to find a pair of very large numbers, at least one of which is transcendental. But the current lack of knowledge about this particular number is even an integer shows just how much progress remains to be made, in my opinion. Any answers that describe techniques that would suffice to solve the problem (perhaps with other, unproven assumptions) would be very welcome.

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    $\begingroup$ This is likely to be open, I think. $\endgroup$ – Qiaochu Yuan Dec 5 '10 at 8:21
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    $\begingroup$ If we don't even know if $\exp(e)$ is algebraic or not... $\endgroup$ – J. M. isn't a mathematician Dec 5 '10 at 9:41
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    $\begingroup$ @Qiaochu Yuan and J.M.: I wasn't aware of that. I knew (vaguely) that there are a lot of open problems in transcendence theory, but I was hoping that just the problem of being an integer was easier. $\endgroup$ – Carl Mummert Dec 5 '10 at 12:56
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    $\begingroup$ It's not particularly feasible; the number of decimal digits is approximately $\log_{10}$ of the number, which is much larger than $10^{80}$, which is supposed to be an estimate for the number of atoms in the observable universe. $\endgroup$ – Carl Mummert Dec 6 '10 at 23:03
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    $\begingroup$ @Carl This is a really nice question. I don't think it needs to be "closed". There are some really innocuous questions running around the site just because the person who asked them never accepted an answer. But I believe that this is a question that deserves to show up from time to time. Who knows, maybe at some point someone will have something important to say about it, if not to answer it for good =) $\endgroup$ – Adrián Barquero Mar 20 '11 at 16:29

The paper Chuangxun Cheng, Brian Dietel, Mathilde Herblot, Jingjing Huang, Holly Krieger, Diego Marques, Jonathan Mason, Martin Mereb, and S. Robert Wilson, Some consequences of Schanuel’s conjecture, Journal of Number Theory 129 (2009) 1464–1467, shows that $e,e^e,e^{e^e},\dots$ is an algebraically independent set, on the assumption of Schanuel's Conjecture. Maybe a close reading of that paper will suggest a way of applying the result to the 79-question.

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    $\begingroup$ Maybe writing to the authors and telling them about this question directly could prove useful. $\endgroup$ – Adrián Barquero Jun 16 '11 at 4:38
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    $\begingroup$ Unfortunately Schanuel's Conjecture (that if $n$ complex numbers are linearly independent over the rationals, there are at most $n$ algebraic dependencies between those $n$ numbers and their exponentials) is completely intractable. $\endgroup$ – HTFB Jun 3 '13 at 9:20

if $e^{e^{e^{79}}}$ is an integer then $e^{e^{e^{e^{79}}}}$ is not an integer (otherwise $e$ would be algebraic). Perhaps your arguments make sense with this number too.

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    $\begingroup$ This is a good point. Unfortunately, for my example I need to know for sure whether it's an integer or not; knowing that at least one of two numbers is not an integer is not as useful. $\endgroup$ – Carl Mummert Jun 16 '11 at 0:14

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