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 -
- 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).
- 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.
$\pi$ 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.