A string of digits that appears infinitely many times in $\pi$

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Anindya Biswas
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A string of digits that appears infinitely many times in $\pi$

Unread post by Anindya Biswas » Mon Feb 15, 2021 10:45 pm

Prove that there exists a $2021$ digits long string of digits that appears infinitely many times in the decimal representation of $\pi$.
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Mehrab4226
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Re: A string of digits that appears infinitely many times in $\pi$

Unread post by Mehrab4226 » Mon Feb 15, 2021 10:51 pm

Easy!!! :). Use induction. This is true for all irrational numbers
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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Re: A string of digits that appears infinitely many times in $\pi$

Unread post by Anindya Biswas » Mon Feb 15, 2021 10:56 pm

Mehrab4226 wrote:
Mon Feb 15, 2021 10:51 pm
Easy!!! :). Use induction. This is true for all irrational numbers
Induction? I was thinking about infinite pigeonhole principle.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Mehrab4226
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Re: A string of digits that appears infinitely many times in $\pi$

Unread post by Mehrab4226 » Mon Feb 15, 2021 11:04 pm

It is easy to see that there are infinitely many one digit strings in $\pi$ since $\pi$ have infinitely many pigeons and only $10$ pigeonholes.
Now, Assume that there is a string $n$ digit long which appears infinitely may times. Now, we make a new string with the previous string and the digit right after that. So we will get an $n+1$ digit string. There are infinitely many $n+1$ digit strings with n of them common. There can be only $10$ different types of $n+1$ strings. Thus by PHP there should be infinitely many $n+1$ digit strings in $\pi$. This is true for any irrational number. $\square$.
I also used PHP in it. But not infinitely many times. Probably 2 times. And using induction gives us a more generalized proof of the problem.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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Mehrab4226
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Re: A string of digits that appears infinitely many times in $\pi$

Unread post by Mehrab4226 » Mon Feb 15, 2021 11:12 pm

PHP may be used to prove the case with 2021 digits. But using induction will make your work way easier. Since you only need to show when the string is 1 digit long!.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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Anindya Biswas
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Re: A string of digits that appears infinitely many times in $\pi$

Unread post by Anindya Biswas » Mon Feb 15, 2021 11:19 pm

Mehrab4226 wrote:
Mon Feb 15, 2021 11:04 pm
It is easy to see that there are infinitely many one digit strings in $\pi$ since $\pi$ have infinitely many pigeons and only $10$ pigeonholes.
Now, Assume that there is a string $n$ digit long which appears infinitely may times. Now, we make a new string with the previous string and the digit right after that. So we will get an $n+1$ digit string. There are infinitely many $n+1$ digit strings with n of them common. There can be only $10$ different types of $n+1$ strings. Thus by PHP there should be infinitely many $n+1$ digit strings in $\pi$. This is true for any irrational number. $\square$.
I also used PHP in it. But not infinitely many times. Probably 2 times. And using induction gives us a more generalized proof of the problem.
Cool, my argument is there are only $10^{2021}$ such strings and they need to cover this infinitely many digits, at least one such string should repeat infinitely times. Now this part is simple, let's ask something big, does there exists one that appears only finitely many times? Can they be known? I don't know.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Mehrab4226
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Re: A string of digits that appears infinitely many times in $\pi$

Unread post by Mehrab4226 » Mon Feb 15, 2021 11:38 pm

Anindya Biswas wrote:
Mon Feb 15, 2021 11:19 pm
Mehrab4226 wrote:
Mon Feb 15, 2021 11:04 pm
It is easy to see that there are infinitely many one digit strings in $\pi$ since $\pi$ have infinitely many pigeons and only $10$ pigeonholes.
Now, Assume that there is a string $n$ digit long which appears infinitely may times. Now, we make a new string with the previous string and the digit right after that. So we will get an $n+1$ digit string. There are infinitely many $n+1$ digit strings with n of them common. There can be only $10$ different types of $n+1$ strings. Thus by PHP there should be infinitely many $n+1$ digit strings in $\pi$. This is true for any irrational number. $\square$.
I also used PHP in it. But not infinitely many times. Probably 2 times. And using induction gives us a more generalized proof of the problem.
Cool, my argument is there are only $10^{2021}$ such strings and they need to cover this infinitely many digits, at least one such string should repeat infinitely times. Now this part is simple, let's ask something big, does there exists one that appears only finitely many times? Can they be known? I don't know.
Your solution is concrete too. Hmm......
If there is a string repeating finitely many times, right?
Then we can prove, it is true for some irrationals not all of them. Because we can make an irrational number by using all strings infinitely many times.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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