# Can the sequence of successive digits of $\pi^{18}$ ever give a prime?

In this question

First $k$ digits of $\pi^n$ and compositeness

it is asked for some $$\ n\$$ giving late or possible never a prime number. A good condidate is $$\ n=18\$$. According to my calculations with PARI/GP, we do not get a prime after more than $$\ 11\ 000$$ digits. Note that the digits before the comma are used as well and we also do not arrive at a prime before reaching the comma.

Is $$\ \lfloor \pi^{18}\cdot 10^k \rfloor\$$ ever prime ?

Heuristically, we can expect that a prime will eventually occur , if we assume that the digits of $$\ \pi^{18}\$$ behave like a pseudo-random-generator. Motivation for the $$\ 18\$$ is that it is the first tough case (see the table in the answer).

• What did I say ? The downvote is there :) Mar 14, 2020 at 13:45
• Is it clear that a (uniformly) randomly generated string yields a prime at some point with probability $1$? Is there an expected density of primes produced in this way?
– lulu
Mar 14, 2020 at 13:55
• @lulu Even if clear if it´s random, primes are not, only in random models about them.
– user757601
Mar 14, 2020 at 13:57
• @lulu This has a point. Even, if the probability is not exactly as the 1/ln(n)-model predicts it is probably large enough to ensure that we must arrive at a prime eventually, if the digits are actually random. Since we have no reason to doubt that the digits of $\pi^{18}$ behave like a random-generator, I highly exect not only one prime, but infinite many. But the smallest could well be too large to be ever detected ! Mar 14, 2020 at 14:05
• In the mean time, I passed $\ 14\ 000\$ digits Mar 14, 2020 at 14:07

## 1 Answer

$$\large \left \lfloor \pi ^{18} \cdot 10^{16718} \right \rfloor$$ is a probable prime.

Pari/GP code:

\p 16800
ispseudoprime(floor(Pi^18*10^16718))
time = 1min, 22,156 ms.
%# = 1

• I see, Peter already found it about an hour before I posted. Mar 14, 2020 at 16:23