Is there a section of mathematics that studies near-integer equations.

Question

When I saw:

$$e^\pi-\pi \approx 20$$

I thought it was pretty cool. And :

$$\pi^3 \approx 31$$

So now the thought comes to me is what positive integer value of $n$ will make the expression:

$$\pi^n$$

As close to a nearest integer as possible. Furthermore is there a section of mathematics devoted to studying these kinds of things?

My guess would be to deal with floor and ceiling functions, but trying to minimize these things seems out of reach.

Answer 1

What positive integer value of $n$ will make the expression $\pi^n$ as close to a nearest integer as possible?

The following PARI/GP line will choose $n=157$ as the best integer exponent up to $1024$.

for(n=1,2^10,if(abs(round(Pi^n)-Pi^n)<1/2^10,print(n)))

You may download the software here: http://pari.math.u-bordeaux.fr/download.html