In the case of $\dfrac{1}{7^{800}}$ it's easy, to find the $2^{2020}$ decimal, but what about the simplest of the irrational numbers.

Question: Do we know how to determine the $2^{2020}$ decimal of $\sqrt{2}$?

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    $\begingroup$ @Qfwfq Newton-Rapson won't finish before the Sun explodes and won't have enough particles in the visible universe for the required memory storage, but otherwise it is fine. A legitimate answer should produce the required digit without computing all the preceding ones and that may be quite tricky since there is no obvious pattern in the digit sequence. I would say the answer is "no" but there is no proof of that either. $\endgroup$ – fedja Feb 23 '20 at 21:58
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    $\begingroup$ according to arXiv:0912.0303 no digit-extraction formula is known for $\sqrt 2$. $\endgroup$ – Carlo Beenakker Feb 23 '20 at 22:11
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    $\begingroup$ This problem has nothing to do with arithmetic geometry. Arithmetic geometry is algebraic geometry over finite fields, local fields, global fields etc. $\endgroup$ – GH from MO Feb 23 '20 at 22:53
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    $\begingroup$ Not voting to reopen because the question seems hopeless, but, of course, the suggestion to send it to MSE (implied by the formal closing reason) is ridiculous. $\endgroup$ – fedja Feb 24 '20 at 0:27
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    $\begingroup$ @IgorKhavkine It’s a completely different algorithm. Basically, you compute the square root by partially summing a power series, but the hard part is to compute the relevant iterated additions and multiplications without storing all the terms (recomputing some of it when needed). The basic idea of the iterated multiplication algorithm is to do it modulo many small primes and then reconstruct it from the Chinese remainder representation, but there is more to it than that. See Hesse, Allender, and Barrington doi.org/10.1016/S0022-0000(02)00025-9 . $\endgroup$ – Emil Jeřábek Feb 24 '20 at 7:59

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