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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 -- 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… ). 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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