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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 $\pi$?

Paul Frost
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J. Mini
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  • You can use use approximations such as ramanajun’s or other basic series expansions of arcsin(x) etc.? – Тyma Gaidash Apr 28 '21 at 18:22
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    I have never heard of anything like this. There are two kinds of methods: ones that must perform iterative computation at full precision, and others that work on a (relatively) short sequence of decimals, but none uses an explicit relation between the decimals. –  Apr 28 '21 at 18:25
  • Using series expansions doesn't directly address the question. It seems the question is asking about the randomness of the digits of pi? – Nicholas Stull Apr 28 '21 at 18:42
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    As a general rule when calculating the digits of a number, you make successive approximations to it. In such algorithms, it is critical to know the $n^{th}$ digit as well as all the digits before it when searching for the next. That said, there is an algorithm that allows you to calculate a given hexidecimal digit of $\pi$ without calculating those before it. It is inefficient for calculating the value of $\pi$ itself, but is used in validating the results of other algorithms. – Paul Sinclair Apr 29 '21 at 03:40
  • Yes @PaulSinclair, this reminds me of strong induction. – Zuter_242 May 11 '21 at 12:40
  • @Zuter_242 - true, but it is a bit like saying "Justin Timberlake reminds me of Albert Einstein, because they are both male and have hair". What the two processes have in common (iterative, dependent on all earlier results) is common to many other regularly used processes as well. – Paul Sinclair May 11 '21 at 15:51
  • Point taken, @PaulSinclair, when I say "this reminds" I mean "these two things happen to have a trait in common". – Zuter_242 May 11 '21 at 17:43
  • The reason your question has garnered little response is that it is not at all clear what you are after. Several of us have taken somewhat random shots at trying to give useful information, but you have not clarified at all what you are after or even offered any response that would let us know if what we said was useful or completely un-related. You would be far better served by providing more information about what you want, than by offering a bounty that is effectively a lottery. – Paul Sinclair May 11 '21 at 23:16

1 Answers1

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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.

Troposphere
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  • This is a convincing heuristic, but I feel that there are some interesting questions raised in what you mean by "in a precise technical sense" - sure, in terms of probability/asymptotic density, the digits are independent. But does that imply the same in the context of computation? It's not clear to me that "hard to guess" (probability) and "hard to calculate" (computation) are necessarily related, but I'm not sure. – Milo Brandt May 09 '21 at 02:44
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    @Milo: Indeed. As the answer points out, the assumed normalcy of $\pi$ doesn't prevent some approximation algorithms from making (at least minimal) use of such knowledge, and for all we know, there could be an even faster algorithm than the currently known ones which could take additional advantage of this knowledge. I mention both sides of the answer because it doesn't seem quite clear which of them the OP is really interested in -- and he might feel it enlightening to see both described and have it pointed out that they don't mean the same! – Troposphere May 09 '21 at 02:50