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How to show $e^{e^{e^{79}}}$ is not an integer

Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it?

Here ${^5\pi}$ means the result of tetration $\underbrace{\pi^{\pi^{\pi^{\pi^\pi}}}}_{5 \text{ times}}$.

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    If you need to ask whether something can be proved, then it is not _obviously_ true. (Except if your question is "do we need to take this as an axiom or does it follow from simpler principles?").2011-12-13
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    Excuse my obvious irony, but I find it ironical that the obviously is obviously ironical.2011-12-13
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    @HenningMakholm: Of course, that is why I put _"obviously"_ in quotes. I just mean that a possible proof that ${^5\pi}$ is not an integer would hardly be a surprising result for anyone, but the converse would really surprise many people.2011-12-13
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    @AndresCaicedo: Do you have a strong reason to believe that both questions will be resolved in the same direction or by similar methods?2011-12-13
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    For some reason, WolframAlpha says it is not an integer: http://www.wolframalpha.com/input/?i=Is+pi%5Epi%5Epi%5Epi%5Epi+an+integer%3F2011-12-13
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    If we use [fraction](http://www.wolframalpha.com/input/?i=Is+pi%5Epi%5Epi%5Epi%5Epi+a+fraction%3F) instead integer the problem looks hard^^2011-12-13
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    @GarouDan: the problem is actually hard even if we use 'integer', because we simply don't know pi to enough places to be able to use interval arithmetic to bound it between two adjacent integers.2011-12-14
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    I don't know how the [IntegerQ](http://www.google.com.br/url?sa=t&rct=j&q=integerq&source=web&cd=1&ved=0CCgQFjAA&url=http%3A%2F%2Freference.wolfram.com%2Fmathematica%2Fref%2FIntegerQ.html&ei=9uznTpWQNYrFtgfx1ZH0CQ&usg=AFQjCNF4tzs8WA9yCYV8bDgHScLVMOmg3Q) Mathematica algorithm works, but if we know this, I think is done.2011-12-14
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    The words "by the way" seem inappropriate here.2011-12-14
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    Four upvotes seems inappropriate!2011-12-14
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    @VladimirReshetnikov, by "converse" I take it you mean "negation".2011-12-14
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    What is the motivation for asking about this particular number?2011-12-14
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    @GerryMyerson: Yes, I meant "negation". Sorry for confusion.2011-12-14
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    @JonasMeyer: This is the smallest number of the form ${^n\pi}$ for which I do not know the answer. Tetration is the first hyperoperator (after addition, multiplication and exponentiation) for which the question is not trivial. And $\pi$ is just a natural example of a transcendental number.2011-12-14
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    @TheChaz What do you mean?2011-12-14
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    I mean that people use votes to indicate something other than "this shows research effort". That's not to say that you won't get an upvote from me (eventually). I DO appreciate your continued interaction with those who would help you.2011-12-14
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    Wolfram Alpha says that $(\phi^5-\tau^5)/\sqrt{5}$, where $\phi = (1+\sqrt{5})/2$ and $\tau = (1-\sqrt{5})/2$, is not an integer: http://www.wolframalpha.com/input/?i=Is+%28%28%281%2Bsqrt%285%29%29%2F2%29^5+-+%28%281-sqrt%285%29%29%2F2%29^5%29%2Fsqrt%285%29+an+integer%3F -- but it clearly is, because it's the fifth Fibonacci number, namely 5. (WA then gives a "decimal approximation" which is 5 followed by a couple thousand zeroes.)2011-12-14
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    @MichaelLugo lol. `IntegerQ[(((1 + Sqrt[5])/2)^5 - ((1 - Sqrt[5])/2)^5)/Sqrt[5]]` doesn't works too. Bad implemented this function.2011-12-14
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    @GarouDan "IntegerQ[expr] returns False unless expr is manifestly an integer (i.e. has head Integer)." http://reference.wolfram.com/mathematica/ref/IntegerQ.html2011-12-14

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