Do All Digit Sequences in Pi Occur Equally Often?

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In summary, the question of whether all digit sequences in Pi occur equally often is a topic of mathematical inquiry. While Pi is believed to be a normal number, which would imply that every finite sequence of digits appears with equal frequency in its decimal expansion, this has not been proven. Current research suggests that certain patterns may emerge while also indicating that the distribution of sequences might not be uniform. The ongoing exploration into the nature of Pi continues to fascinate mathematicians and enthusiasts alike.
  • #1
saddlestone-man
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TL;DR Summary
Is any sequence of digits equally likely in the value of Pi?
As the value of Pi is taken to more decimal places, does any sequence of digits become equally likely?

I'm thinking of sequences like ...123456789.... ...333333... and so on.

best regards ... Stef
 
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  • #2
saddlestone-man said:
TL;DR Summary: Is any sequence of digits equally likely in the value of Pi?

As the value of Pi is taken to more decimal places, does any sequence of digits become equally likely?

I'm thinking of sequences like ...123456789.... ...333333... and so on.

best regards ... Stef
It is supposed that ##\pi## is a normal number, where the digits are uniformly distributed. This is, however, unproven.

https://en.wikipedia.org/wiki/Normal_number
 
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  • #3
PeroK said:
It is supposed that ##\pi## is a normal number, where the digits are uniformly distributed. This is, however, unproven.

https://en.wikipedia.org/wiki/Normal_number
Not only that, pi has been calculated out to an enormous number of decimal places and no special repetition has been observed. As @PeroK said, nothing is proven so It may well happen at some point but there's no reason to believe that it will and the evidence so far is against it.

1711633044825.png
 
  • #4
Carl Sagan used something like that as a plot device in his novel Contact. The protagonist, Ellie Arroway finds proof of her trip only in the digits of PI in some base where they discover a hidden message in the image of a circle embedded in the sequence.

Pi is a transcendental number and has a non-repeating decimal sequence. It is assumed to be evenly distributed across all the digits but there's no proof as yet.

https://en.wikipedia.org/wiki/Pi

I imagine it would be a fun exercise to explore.
 
  • #5
phinds said:
View attachment 342466
I would assume that in the first 105 trillion known digits with a hypothesized uniform distribution, a short sequence like '123456789' or '333333333' already appears somewhere in the 105 trillion. I haven't calculated the probability, assuming a uniform distribution.
 
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  • #6
Many thanks for the answers.

I wondered that there may be some sequences of digits (say 1,000 repetitions of the same digit, or something like 123123123123123123 ....) which would indicate that the value had converged, which of course in the case of Pi is impossible.
 
  • #7
saddlestone-man said:
Many thanks for the answers.

I wondered that there may be some sequences of digits (say 1,000 repetitions of the same digit, or something like 123123123123123123 ....) which would indicate that the value had converged, which of course in the case of Pi is impossible.
By 'indicate that the value had converged', do you mean 'fool you into thinking that the value had converged'?
 
  • #8
I think what I'm saying is .... say you had two long sequences of numbers, one of which was extracted from the value of Pi and the other which was the tail end of a large long division (whose answer did not have an infinite number of digits): would you be able to tell the difference?
 
  • #9
No. Any finite sequence of digits within ##\pi## could be from the division of the integer represented by that sequence by the appropriate power of 10.
 
  • #10
Since this is in the Statistics section, maybe you can look up tabulated data and do a ##\chi^2## goodness of fit for the uniform distribution on ##\{0,1,2,..., 9\}##?
 

FAQ: Do All Digit Sequences in Pi Occur Equally Often?

What does it mean for all digit sequences to occur equally often in Pi?

For all digit sequences to occur equally often in Pi, it means that Pi would have the property of being a "normal number." In a normal number, every possible sequence of digits of any given length appears with equal frequency in the infinite decimal expansion of the number.

Is Pi a normal number?

As of now, it has not been proven whether Pi is a normal number. Extensive computational analysis has shown that Pi's digits appear to be uniformly distributed, but a formal proof of normality is still lacking.

How are the digits of Pi generated?

The digits of Pi are generated using various algorithms that compute its decimal expansion to an arbitrary number of digits. Some common algorithms include the Bailey-Borwein-Plouffe (BBP) formula and the Gauss-Legendre algorithm.

What is the significance of knowing if Pi is normal?

Determining whether Pi is normal has implications in number theory and randomness. If Pi is normal, it would confirm that its digits are uniformly distributed, which can be important for applications in statistics, cryptography, and pseudorandom number generation.

Have any other numbers been proven to be normal?

Yes, there are numbers that have been proven to be normal. For example, the binary Champernowne constant, which is formed by concatenating the binary representations of the natural numbers, is known to be normal in base 2. However, proving normality for naturally occurring constants like Pi and e remains an open challenge.

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