A string of digits that appears infinitely many times in $\pi$
- Anindya Biswas
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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 $\pi$
Easy!!! . Use induction. This is true for all irrational numbers
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- Anindya Biswas
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Re: A string of digits that appears infinitely many times in $\pi$
Induction? I was thinking about infinite pigeonhole principle.Mehrab4226 wrote: ↑Mon Feb 15, 2021 10:51 pmEasy!!! . Use induction. This is true for all irrational numbers
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- Mehrab4226
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Re: A string of digits that appears infinitely many times in $\pi$
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
-Henri Poincaré
- Mehrab4226
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- Joined:Sat Jan 11, 2020 1:38 pm
- Location:Dhaka, Bangladesh
Re: A string of digits that appears infinitely many times in $\pi$
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 $\pi$
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.Mehrab4226 wrote: ↑Mon Feb 15, 2021 11:04 pmI 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.
"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 $\pi$
Your solution is concrete too. Hmm......Anindya Biswas wrote: ↑Mon Feb 15, 2021 11:19 pmCool, 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.Mehrab4226 wrote: ↑Mon Feb 15, 2021 11:04 pmI 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.
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
-Henri Poincaré