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Is it possible to predict next digit in $\pi$ using $N$ previous digits, so on and so forth? Or is this impossible because it's irrational?

Basic assumption is that the person doesn't know a particular sequence of digits is $\pi$.

By predict I mean either of the following -

  1. To have a mathematical equation in which $N+1$th digit can be expressed as a function of previous $N$ digits, without using the value $\pi$ (that would be cheating I guess).
  2. To have a machine learning algorithm that is trained on say 1000 instances of $N+1$ digits, and is able to predict with reasonable accuracy what the next digit in a given test sequence of $N$ digits be.
user13107
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    I can predict the $n$-th digit of $\pi$ with 0 previous digits! :) – Bill Cook Jan 20 '14 at 02:43
  • You should really know that $\pi$ is of course irrational. If the number is rational, then of course we can use the previous digits. – NasuSama Jan 20 '14 at 02:43
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    @NasuSama, how does knowing the previous digits help predicting any further digits, even in the case of rational numbers? The period may be way larger than what has been seen so far. It may not even having begun repeating at all. – lhf Jan 20 '14 at 03:05
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    You can compute hexadecimal digits of $\pi$ with no prior knowledge. – pjs36 Jan 20 '14 at 03:05
  • What does "no prior knowledge" mean, @pjs36 - you can also compute the $n$th decimal digit without prior knowledge. (Yes, I know the base $8$ result, just being pedantic in response to your pendatry.) \ – Thomas Andrews Jan 20 '14 at 03:12
  • I think, at heart, the problem is, what does "predict" mean? And is $N$ fixed, or do you mean predict the $N+1$th digit based on the first $N$ digits? – Thomas Andrews Jan 20 '14 at 03:16

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$\pi$ is believed (though not proven) to be a normal number. If it is, then knowing $N$ digits starting at an unknown location gives no information about the next digit, since all $(N+1)$-digit substrings have the same natural density. Conversely, if $\pi$ is not normal, then there are some strings of digits that give nonzero information about the next digit in the sequence.

mjqxxxx
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