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

Let $$\text{fd}(k,x)$$ be first $$k$$ digits of some real number $$x$$.

For $$\pi=x$$ we have the sequence $$,3,31,314,3141,3141,31415,...$$ (in base $$10$$)

For $$\pi^2=x$$ we have $$9,98,986,9869,...$$ (in base $$10$$)

And so on.

I came to an idea of thinking does there exists $$m \in \mathbb N$$ such that $$k \to \text{fd}(k,\pi^m)$$ are all composite numbers?

This seems to be highly unlikely, and I do not know how to provide a proof.

This question on MO.

• @Peter This is about all being composites, not about finite sequences of primes.
– user757601
Mar 14, 2020 at 12:44
• Sorry, but what do you think about Haran's suggestion that we should look at the fractional part ? Mar 14, 2020 at 12:45
• Heuristically, a prime should occur at some point, but I think we can find very tough cases. Mar 14, 2020 at 12:45
• @Peter That´s just a different question, very similar.
– user757601
Mar 14, 2020 at 12:45
• @Peter How much tough?
– user757601
Mar 14, 2020 at 12:46

If we consider all digits (as suggested in the question), the first tough case is for $$\ m=18\$$. The search is still in progress. I passed $$\ 9\ 000\$$ digits without finding a prime.

The smallest $$\ n\$$ such that $$\lfloor x\cdot 10^n \rfloor$$ is prime can be seen in the following table. A positive entry means that this number of digits after the comma is needed. A non-positive number indicates that we get a prime already before the comma is reached :

1  0
2  106
3  -1
4  0
5  -2
6  1
7  -3
8  2
9  -4
10  27
11  -5
12  0
13  -6
14  -1
15  -7
16  6
17  -8

• A probable prime occured for $\ n=16\ 718\$ Mar 14, 2020 at 15:04
• The next possibly tough case is $\ \pi^{237}\$ which passed $\ 6\ 000\$ digits. Mar 14, 2020 at 17:05