What is the largest number of consecutive repeating digits in pi?

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Discussion Overview

The discussion revolves around the existence of long sequences of repeating digits in the decimal expansion of pi, specifically whether a string of 100 consecutive 2's has been found. Participants explore concepts related to normal numbers and the implications of digit distribution in pi.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that pi has been calculated to over 3 trillion decimal places and questions if a string of 100 2's exists, suggesting it should be possible.
  • Another participant counters that while such a sequence might exist, there is no guarantee it must appear, referencing the belief that the digits of pi are "random" but stating this has not been proven.
  • A third participant introduces the concept of a normal number, explaining that in a normal number, every finite sequence of digits occurs with equal frequency, but emphasizes that pi has not been shown to be normal, leaving the existence of a trillion consecutive 2's uncertain.
  • Several participants express curiosity about how one might prove a number is normal, with references to specific proofs and the challenges involved in demonstrating normality.
  • One participant discusses the Copeland-Erdos constant as an example of a normal number, noting its counter-intuitive nature regarding the distribution of digits among prime numbers.
  • Counter-intuitive facts about normal numbers are shared, including that almost all real numbers are normal and that proving a number's normality is complex.

Areas of Agreement / Disagreement

Participants express differing views on the existence of long sequences of digits in pi, with some suggesting it should be possible while others argue there is no certainty. The discussion on normal numbers also reveals a lack of consensus on the implications for pi.

Contextual Notes

The discussion highlights the unresolved nature of pi's normality and the complexities involved in proving such properties, indicating limitations in current understanding.

MrModesty
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As far as I can tell from googling, pi has been calculated to over 3 trillion decimal places. I'm curious whether a string of 100 2's has been found.

It has to happen, right? As I understand it, we should be able to find a string of 3 trillion consecutive 2's.

Am I wrong in my thinking?
 
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We certainly might be able to find such a sequence but I can see no reason to think that there must be such a sequence. If, as many people believe, but I don't think has been proven, the digits of [itex]\pi[/itex] are "random" (every digit, every sequence of digits is equally likely to appear) it still does not follow that a given sequence will appear.
 
The notion that you're referring to is that of a normal number. A number is called normal if every finite sequence of digits occurs thesame number of times as every other finite sequence of digits. Thus in a normal number, there are as much numbers 2 as numbers 3, there are as much numbers 33 as 13.
So, in particular, in a normal number, you would expect that once in a time there would be a trillion 2's after each other.

The thing is however, that pi hasn't yet been shown to be normal. So we don't really know if there is a sequence with a trillion 2's...

A number that IS normal is Champernowne's number, this is:

0.123456789101112131415161718192021222324...

Thus just concatenate all numbers. If you concatenate all PRIME numbers, then you also get a normal number:

0.23571113171923...

This is called the Copeland-Erdos constant.

However, as you might have guessed, showing that a number is normal is very hard...
 
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Out of curiosity, how would one go about proving a number is normal?
 
micromass said:
If you concatenate all PRIME numbers, then you also get a normal number:

0.23571113171923...

This is called the Copeland-Erdos constant.

That seems counter-intuitive. Since every prime number (except two of them) ends in 1, 3, 7 or 9, one would feel the distribution was going to be biased - unless this effect washes out faster than something else, as the primes get bigger and contain more digits.

I suppose that just illustrate the fact that
However, as you might have guessed, showing that a number is normal is very hard...
 
jhae2.718 said:
Out of curiosity, how would one go about proving a number is normal?

A proof is found in http://www.mth.uea.ac.uk/~h720/teaching/dynamicalsystems/champernowne1933.pdf but be warned, it's not exactly easy reading...
 
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micromass said:
A proof is found in http://www.mth.uea.ac.uk/~h720/teaching/dynamicalsystems/champernowne1933.pdf but be warned, it's not exactly easy reading...

Thanks.
 
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AlephZero said:
That seems counter-intuitive. Since every prime number (except two of them) ends in 1, 3, 7 or 9, one would feel the distribution was going to be biased - unless this effect washes out faster than something else, as the primes get bigger and contain more digits.

I suppose that just illustrate the fact that

I agree 100% with you, it is very counter-intuitive. In fact, if it wasn't proven, then I would have claimed that it wasn't true!

Some other counter-intuitive facts:
- almost all real numbers are normal (in the sense that the non-normal numbers have measure 0). Still it is very hard to come up with numbers which are normal.
- every real number is the product of two normal numbers
- multiplication by a nonzero rational number doesn't change the normalness of a number
 

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