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Mehrab4226 wrote: ↑

Mon Feb 15, 2021 10:51 pm

Easy!!! . Use induction. This is true for all irrational numbers

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.

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.