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I want to compute last few digts (as much as possible ) of the following number $$ N:=n^{n^{n^{\cdot^{\cdot^{\cdot^n}}}}}\!\!\!\hspace{5 mm}\mbox{ if there are $k$ many $n$'s in the expression and $n\in\mathbb{N}$ }$$ I have seen many particular cases of this problem. I think for odd $n$ the units digit is $n^3\mbox{ mod } 10 $ and for even $n$ the units digit is 6, for all $k\geq 3$ . How much can we say about the other digits ?

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    Unless you'll write out the brackets, $N$ is not clearly defined.2012-07-03
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    @penartur I dont see any problem with the definition of $N$2012-07-03
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    $3^{(3^3)} = 3^{27} \ne 3^9 = 3^{3 \times 3} = (3^3)^3$. The operation of raising to the power is not associative.2012-07-03
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    Whilst this is true, I think the usual convention is that the brackets start from the top. So $x^{y^z}$ becomes $x^{(y^z)}$2012-07-03
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    By $n^{n^{n^n}}$ I mean $n^{(n^{(n^n)})}$ and I think this is usually meant by what I wrote2012-07-03
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    You can write it using Knuth's up-arrow notation as $n\uparrow\uparrow k$, that way there's no ambiguity. ;)2012-07-03
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    Also, you say it's $n^3$ -- where is $k$? it is easily not true for e.g. $k=0$, or if you want $k\neq 0$, then $n=k=2$. You also need to specify what kind of answer you're expecting. There is a way to calculate all the digits: just calculate the entire expression! But that's not what you want, is it?2012-07-03
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    @tomasz that is true for all $k\geq 4$ I have edited it. Thanks2012-07-03
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    http://math.stackexchange.com/a/162608/26068 This may be helpful2012-07-03
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    Basically i want know whether there are some good techniques to predict the digits without calculating the whole expression ( and i think this is not possible even for n=2 and k=10 within the lifetime of universe).2012-07-03
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    This is known as tetration and there's a forum devoted to it at http://math.eretrandre.org/tetrationforum/index.php.2012-07-03
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    I realized that for even $n$ and $k\geq 3$ the units digit is different, it is 6. I edited this.2012-07-03
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    See http://oeis.org/A183613 and references there.2012-07-03

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