Patterns in Pi

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Main Question or Discussion Point

Has anyone every discovered any patterns in pi such as...every 97th digit is a 3? or something similar?

I know there is no repeatable pattern of digits but is there anything that precludes other patterns such as the above?

tex

ProfuselyQuarky
Gold Member
Pi is irrational. There is simply no pattern and it's not like anything prevents a pattern from occurring. Pi is just pi.

mfb
Mentor
Has anyone every discovered any patterns in pi such as...every 97th digit is a 3? or something similar?
No. It is expected that no such pattern exists ("normal number"), and billions to trillions of digits have been analyzed to look for patterns (for a huge amount of things you could call "pattern"), but there is no proof that no such pattern exists.
Pi is irrational. There is simply no pattern and it's not like anything prevents a pattern from occurring. Pi is just pi.
There are irrational numbers with patterns like the one described by thetexan.

ProfuselyQuarky
Gold Member
There are irrational numbers with patterns like the one described by thetexan.
You are right.

Well, that brings up this question.

Let's say they did find a pattern in an irrational number...every 105th digit is a 7. What would that imply? Would it still be irrational? Would we be able to figure out why the pattern exists?

tex

fresh_42
Mentor
Well, that brings up this question.

Let's say they did find a pattern in an irrational number...every 105th digit is a 7. What would that imply? Would it still be irrational? Would we be able to figure out why the pattern exists?

tex
If we include a 7 at every 107th digit of $\pi$ I would bet the new number will still be transcendental. There are still 106 digits where anything can happen.

mfb
Mentor
There are irrational numbers where every 105th digit is 7. That is perfectly fine - as long as the number does not have a fixed sequence (and nothing else after some point) repeated to infinity in its decimal expansion it is irrational.

As an example, if you replace every 105th digit in the decimal expansion of pi by 7, you get an irrational number with that pattern.
Would we be able to figure out why the pattern exists?
What does "why" even mean? Why is pi larger than 3?

It would be extremely odd, because such a pattern would depend on the base we use - we use base 10, in binary, ternary or whatever else you would not have that pattern.