An unknown pattern in $π^k$.

Question

While trying to find a pattern in $π$, by taking the idea from $n$ such that the digits immediately after the decimal point of $\pi^n$ give $n$ again

By using wolfram alpha calculator first I am writing the examples of my invesgation

π³ = 31.00627668....

π⁵ = 306.0196847....

π⁹ = 29809.09933344....

π¹⁰ = 93648.0474760830....

π¹¹ = 294204.01797389059....

π¹⁸ = 888582403.07126338067....

As we can see that in each example after a decimal place there is '0'. But I don't know any math programming language to calculate further more values.

Now the question is that is there are infinite values 'k' such that:

π^k = .....abcdefhg.0wpywqx.... (in which after decimal place '0' will be there)

Any hint or solution is welcome. Thanks.

Answer 1

Here's a histogram of the first digit (to the right of ".") values for $\pi^n$ for $1 \leq n \leq 500$: Clearly the results are random and there is nothing special about the value $0$ in that digit position:

Here's the Mathematica code for generating the histogram:

piList = Table[N[π^n, 500], {n, 500}];
digitList = IntegerPart /@ (10 (FractionalPart /@ piList));
Histogram[digitList]