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Can we ever know the sum of the first $10^{10^{100}}$ digits of $\pi$?

Can we calgulate the $n$th digit of $\pi$ when the Kolmogorov-complexity of $n$ is larger than the complexity of the calculating system?

Example: We cannot calculate digit number 314159265358 of $\pi$ on a typical pocket calculator.

EDIT: As Goldstern remarks, we cannot write down all the digits simultaneously. But can we know every desired shorter string of digits, for instance the first billion following the digit number $10^{{10}^{90}}$ and then note their sum, and so on?

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I don't see a mathematical question here, but perhaps I'm just blind. –  Alex Becker Feb 2 '13 at 12:47
The second question is along the lines of scottaaronson.com/blog/?p=46 -- we take a number which is so large as to be uncomputable with and ask whether we can compute some very trivial aspect of it. (See also math.stackexchange.com/questions/271834/… ). No one seems to have made much progress on either of those questions. –  David Speyer Feb 2 '13 at 12:53
The sum of the first $10^{10^{100}}$ digits of $\pi$ has most likely more than $10^{80}$ digits, so where would you write it down? –  Goldstern Feb 2 '13 at 13:50
That is just implied by my question: If matheamtics is primarily communication between inhabitants of the universe, does this sum belong to mathematics? Or is the "universe of discourse" larger? –  Rhett Butler Feb 2 '13 at 15:05
Given your name, perhaps you will understand when I answer your questions, "Frankly, my dear, I don't give a damn." –  Gerry Myerson Feb 2 '13 at 23:09
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