Does knowing the nth digit of $\pi$ help in finding the next digit?

Question

Obviously, we can generate as many digits of $\pi$ as we please, but is there any way to make use of knowing digit $n$ when seeking next digit of $\pi$?

Answer 1

That depends, perhaps, on which kinds of "help" you're willing to accept.

$\pi$ is widely expected to be normal in all bases. If this is true, an arguable answer to your question would be "no": In a precise technical sense, knowing one digit of $\pi$ tells you nothing about what the next digit is. If you pick a digit position at random, every digit is equally likely to be there no matter what the preceding digit is.

(A caveat here is that nobody has been able to prove that $\pi$ is normal; we mostly believe it is because nobody has been able to suggest a good reason it wouldn't be normal either, and the trillions of digits that have been computed so far sure look like it's normal in base 10).

On the other hand, some possible algorithms for approximating $\pi$ can be sped up very slightly by knowing any one of the previous digits. For example, suppose you compute $\pi$ by bisection, based on testing whether candidates are greater or smaller than $\pi$ (say, by computing $\cos\frac x2$ for each candidate $x$ using the power series), knowing a digit in advance can allow you to narrow the candidate interval to get the next digit right without any work once during the entire computation. However, the most efficient known methods for computing $\pi$ are not directly amenable to this optimization anyway -- choosing a method that can make use of the next-to-last digit would be a net loss in efficiency.