Digits of $\pi$ forming primes? Let $f(n)$ be the first $n$ digits of $\pi$ . How many times is $f(n)$ prime on the interval $[1, k]$ ? Are there infinitely many prime $f(n)$'s on $[1, \infty)$?
I know this is probably be a very hard question and most likely an open problem, but I am just curious.
 A: Nothing is known for sure, because not much is known about the digit distribution of $\pi$. For instance it is not known known whether $\pi$ is a normal number, but it is (weakly) believed to be, because $\pi$ has no reason not to be (and tests on the first few millions of digits don't say anything to the contrary), and it is known at almost all real numbers are normal. So let's assume that the digits of $\pi$ do indeed appear uniformly at random, and see what that might imply.
There is a popular heuristic, based on the prime number theorem, that a random number with $n$ digits has "probability" $\frac{1}{n \ln 10}$ of being prime. So, by this heuristic, the expected number of times $f(n)$ — the number formed by the first $n$ digits of $\pi$ -- is prime, for $1 \le n \le k$, is
$$\sum_{n=1}^{k} \frac{1}{n \ln 10} = \frac{H_k}{\ln 10},$$
where $H_k$ is the $k$th harmonic number. As $H_k \sim \ln k$, which (very slowly) goes to $\infty$ as $k \to \infty$, this means that yes, we do expect (based on heuristic) that there are infinitely many $n$ such that $f(n)$ is prime.
Actual calculation has revealed (see OEIS A060421 and OEIS A005042) that $f(n)$ is prime for $n$ in the list (so far)
$$1, 2, 6, 38, 16208, 47577, 78073,$$
the corresponding primes being (listing only the first four)
$$3, 31, 314159, 31415926535897932384626433832795028841, \dots$$
This is roughly consistent with the heuristics, though we found primes slightly earlier. See this page by Kevin Brown for discussion.
