# Irrationality of powers of $\pi$

Everyone knows that $\pi$ is an irrational number, and one can refer to this page for the proof that $\pi^{2}$ is also irrational.

What about the highers powers of $\pi$, meaning is $\pi^{n}$ irrational for all $n \in \mathbb{N}$ or does there exists a $m \in \mathbb{N}$ when $\pi^{m}$ is rational.

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Not only does everyone know that $\pi$ is irrational, but everyone also knows that $\pi$ is transcendental :-) – Robin Chapman Aug 12 '10 at 20:43
@Robin Chapman: Ok. Agreed. Ivan Niven's proof is awesome for the irrationality of $\pi$. – anonymous Aug 12 '10 at 20:45
Does anyone have a link to a proof of transcendentality for completeness? – Casebash Aug 12 '10 at 21:49
The sketch of it at least: if $\pi$ were algebraic, Lindemann-Weierstrass would imply $\exp(2\pi i)$ is transcendental (proving e is transcendental is another story)... and you can fill in the rest. – J. M. Aug 12 '10 at 22:25

If $\pi^{n}$ was rational, then $\pi$ would not be transcendental, as it would be the root of $ax^{n}-b = 0$ for some integers $a,b$.
@Moron: Ok, then if thats the case then why have they proved the seperate case of $\pi^{2}$ being irrational at planet math. They could have used your argument. – anonymous Aug 12 '10 at 20:54
There's a (relatively) simple proof that just happens to work for $\pi^2$ (and from which the $\pi$ case is an immediate corollary). – Robin Chapman Aug 12 '10 at 20:57