Can the First p Digits of Pi Repeat Twice?

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Discussion Overview

The discussion centers around the possibility of the first p digits of pi repeating themselves, specifically whether the first p/2 digits can occur twice in succession. Participants explore the implications of pi's properties as an irrational number and the concept of normality in number theory.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions if the first p digits of pi can be the first p/2 digits repeated, providing an example for clarity.
  • Another participant suggests that while it's conceivable for some irrational numbers to have repeating digits, this does not hold true for pi based on calculations up to a trillion digits.
  • A participant mentions that while no patterns have been found in pi's digits, there is no proof that each digit occurs infinitely in its decimal expansion.
  • Some participants express skepticism about the likelihood of finding such repetitions in pi, suggesting it is possible but very unlikely.
  • There is a discussion about the concept of periodicity in pi, with clarifications that since pi is not rational, it cannot exhibit infinite periodicity.
  • One participant introduces the assumption of normality for pi, proposing that this could allow for a probability calculation regarding the occurrence of such repetitions.

Areas of Agreement / Disagreement

Participants generally agree that while the repetition of digits is theoretically possible, it is considered very unlikely for pi specifically. There is no consensus on the existence of evidence for such occurrences in pi's decimal expansion.

Contextual Notes

Limitations include the lack of proofs regarding the distribution of digits in pi and the unresolved nature of its normality. The discussion also reflects varying interpretations of periodicity and its implications for irrational numbers.

Who May Find This Useful

Readers interested in number theory, the properties of irrational numbers, and the mathematical exploration of pi may find this discussion relevant.

LeBrad
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If I take the first p digits of pi, is it possible that what I will have is the first p/2 digits repeated twice?

For example, suppose pi = 3.14153141529485729487... and I took the first 10 digits and got 3141531415 which is 31415 repeated twice. Can this happen?
 
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It's conceivable that there exists irrational numbers where the first n digits in the decimal expansion repeat themselves and then after 2n digits, it goes non-repeating, but that's not true with pi, at least to a trillion digits.

/edit

Infact, it's not just conceivable, it's pretty obvious!

[tex]\sqrt{2} = 1.4 \ldots[/tex] Therefore [tex]0.123123 + \frac{\sqrt{2}}{10^{7}}[/tex] is irrational, but has the first three digits repeat.
 
Last edited:
AlphaNumeric said:
It's conceivable that there exists irrational numbers where the first n digits in the decimal expansion repeat themselves and then after 2n digits, it goes non-repeating, but that's not true with pi, at least to a trillion digits.

/edit

Infact, it's not just conceivable, it's pretty obvious!

[tex]\sqrt{2} = 1.4 \ldots[/tex] Therefore [tex]0.123123 + \frac{\sqrt{2}}{10^{7}}[/tex] is irrational, but has the first three digits repeat.

Yeah, I knew it wasn't true for arbitrary irrationals because the first example I thought of was (101010+pi) which is irrational and repeats 10 to start. I guess I was looking for existence of evidence that it happens in pi or if there's some way to prove that this never happens for some number.
 
Pi can been calculated to i think its 14 trillion digits? No pattern found, but there are yet no proofs as to why, or even that each digit occurs infinity in the decimal expansion...
 
It is quite possible that for some n, the second n digits in [itex]\pi[/itex] are just the first n repeated but, if so, it must be some exceptionally large n. No one knows for sure but I would consider it unlikely.

I notice you titled this "periodicity of [itex]\pi[/itex]" which implies a block of digits repeating over and over again, forever. Since [itex]\pi[/itex] is NOT rational, that certainly cannot happen.
 
HallsofIvy said:
I notice you titled this "periodicity of [itex]\pi[/itex]" which implies a block of digits repeating over and over again, forever. Since [itex]\pi[/itex] is NOT rational, that certainly cannot happen.

By "finite periodicity" I meant that the first n numbers are repeated a finite number of times, so it may appear periodic if you only look at the first k*n digits.
 
Ah. In that case, as everyone else has told you- possible, unknown, very unlikely! (Unlikely that the first n numbers repeat k times for some n, k. Highly likely that some set of n numbers repeats k times in a row but still unknown.)
 
If I take the first p digits of pi, is it possible that what I will have is the first p/2 digits repeated twice?
With the assumption of normality of pi, can we calculate the probability that this statement is true for some p?
I think for a normal number the probability is less than 1/9.
 

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