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I have heard that it is unknown whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer. How can this be? $\pi$ is known to many digits and it seems like only a matter of time on a computer to find the integer part or to find that there must be a decimal part to the number.

What makes it difficult to determine? I know that it is overwhelmingly likely that the constant is irrational but I am interested in why it is hard to show.

Another aspect of the question could involve answering why there is no proof that for $\alpha$ transcendental $\Large \alpha^{\alpha^{\alpha^\alpha}}$ is not an integer. I am not naive enough to think such a proof would be easy but I also don't know too much about it to know why it would be hard.

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    $\begingroup$ How hard is it to figure out if an electron is a particle or a wave? I mean, it's just an electron. And how hard is it to figure out who killed JFK? There were only finitely many people in Dallas that day. And how hard is it to find out what gene causes what? There are only so few of them. $\endgroup$ – Asaf Karagila♦ Jul 1 '14 at 17:11
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    $\begingroup$ I once heard this quote: "There only are two kinds of problems: the trivial ones, and the ones you can't solve". $\endgroup$ – Ivo Terek Jul 1 '14 at 17:16
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    $\begingroup$ Since $3$ is a lower bound for $\pi$, a lower bound for $\pi^{\pi^{\pi^{\pi}}}$ is given by $3^{3^{3^3}} = 3^{3^{27}} = 3^{7625597484987}$. That's kind of a big(ish) number. $\endgroup$ – Dustan Levenstein Jul 1 '14 at 17:19
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    $\begingroup$ The likely issue is the size of the number rather than any technical difficulty with the proof - if the number is (as is likely) not an integer, it should be possible to bound it away from an integer using sufficiently accurate estimates at each stage. $\endgroup$ – Mark Bennet Jul 1 '14 at 17:21
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    $\begingroup$ @AlonsodelArte: It says the same about $\pi^{\pi^{\pi^\pi}}$ but I doubt it has proved either. $\endgroup$ – Charles Jul 1 '14 at 17:33
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That number has 666262452970848504 decimal places, so to determine if it's an integer you'd have to compute it with that precision. But this would take 270,000 TB, and we don't have many hard drives that large.

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  • $\begingroup$ How much storage would be needed to show that it has a fractional part? (assuming it isn't a near integer) $\endgroup$ – user157227 Jul 1 '14 at 17:22
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    $\begingroup$ You need to compute it to infinite precision if it actually is an integer $\endgroup$ – TROLLHUNTER Jul 1 '14 at 17:23
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    $\begingroup$ @user157227: About 276659495136553954 bytes should suffice. $\endgroup$ – Charles Jul 1 '14 at 17:26
  • $\begingroup$ @user1708: Right -- but the question assumed "it isn't a near integer". $\endgroup$ – Charles Jul 1 '14 at 17:31
  • $\begingroup$ How many decimal places of pi do we have to know to calculate the number within, say .25? Then if the answer is not in the range of x.25 through x.75 (as it has a 50% chance of not being) how many decimal places further to get it within 1/8? $\endgroup$ – fleablood Feb 11 '16 at 2:19
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it is not difficult. The only problem seems to be is that it is to large number to present here all $10^{18}$ digits and the first digit after delimiter

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    $\begingroup$ It may be really difficult if it IS an integer $\endgroup$ – TROLLHUNTER Jul 1 '14 at 17:23
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    $\begingroup$ If you did get the first 1000 digits after the decimal point and they were all $0$, then the question of whether it's an integer would be undecided: maybe the 1001th digit would be $1$. ${}\qquad{}$ $\endgroup$ – Michael Hardy Jul 1 '14 at 17:35
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    $\begingroup$ agree with Michael Hardy, but don't believe that even first 2 are 0. The chances are 1 to 100, I would bet on such condition :) $\endgroup$ – user56396 Jul 1 '14 at 17:47

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