Calculate the exact value $floor({\pi}^{k})$ without actually calculating the huge value of ${\pi}^{k}$.

aangenaamyj
2022-07-08
Answered

Calculate the exact value $floor({\pi}^{k})$ without actually calculating the huge value of ${\pi}^{k}$.

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lofoptiformfp

Answered 2022-07-09
Author has **16** answers

The only way I know of to speed up exponentiation modulo an integer $n$ (compared to general exponentiation) is using the fact that everything is integers to reduce the intermediate results modulo $n$. As the intermediate results in your case aren't integers, you can't do that.

The only transformation I can see of your expression is:

$\lfloor {x}^{k}\rfloor \phantom{\rule{1em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}n)=\lfloor {x}^{k}\phantom{\rule{1em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}n)\rfloor $

and that doesn't make the computation any easier.

The only transformation I can see of your expression is:

$\lfloor {x}^{k}\rfloor \phantom{\rule{1em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}n)=\lfloor {x}^{k}\phantom{\rule{1em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}n)\rfloor $

and that doesn't make the computation any easier.

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