# Is there a positive integer $n$ such that $\lfloor 10^n\pi\rfloor$ is palindromic? [duplicate]

$$\pi$$ is irrational, and of course its digits are not a repeating sequence. However I cannot wrap my head around the following...

Is there a digit $$x$$ in $$\pi$$ such that the digits before are repeated in reverse order after that digit? $$\pi = 3.14159\ldots zyxxyz \ldots 951413 [\ldots]$$

In other words, is there a positive integer $$n$$ such that $$\lfloor 10^n\pi\rfloor$$ is palindromic?

$$\pi$$ has infinite digits, but it is unknown whether any sequence of digits will appear at some point. Moreover, this isn't just a requirement for $$n$$ digits appearing at some position, but for a sequence of $$n$$ digits has to appear at position $$n$$.

• "Pi has infinite digits, so any sequence of digits will appear at some point." Why is that? Commented Aug 5, 2020 at 16:54
• Regarding what @Randall said (I was about to comment myself, then saw his/her comment), note that the digit $2$ does not appear in the following irrational number: $0.101001000100001000001\ldots$ (pattern of $1$'s and $0$'s continues in the suggested pattern) Commented Aug 5, 2020 at 16:57
• You can restate your question as: "Is there a positive integer $n$ such that $\lfloor 10^n\pi\rfloor$ is palindromic?" ... The answer to this is presumably independent of whether "any sequence of digits" appears in $\pi$, so this is a perfectly reasonable stand-alone question. (So, I don't believe this question should have been closed as a duplicate.) In any case, I'd be pretty surprised if the answer were "Yes".
– Blue
Commented Aug 5, 2020 at 17:07
• For those interested, I collected together a lot of Mathematics Stack Exchange questions (made before 9 April 2018) relating to digits of $\pi$ (among other related topics) in my answer to Normal Numbers as members of a larger set? Commented Aug 5, 2020 at 17:12
• A heuristic argument, assuming $\pi$ has "random" digits: since $\sum_{n\ge1}10^{-n/2}=\frac{1}{10-\sqrt{10}}\ll1$, (i) there probably is no such $n$ & (ii) there almost certainly are no more than a few such $n$.
– J.G.
Commented Aug 5, 2020 at 17:56