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I can't find references but I know it has been shown that if $\zeta(3)/\pi^3=a/b$ is rational then $a$ and $b$ are enormous.

EDIT: I found a reference, but not in a formal publication. At http://tech.groups.yahoo.com/group/primenumbers/message/22659?threaded=1&p=2 it says,

"Re: Numerology about the Apery Constant $\zeta(3)$

"I also attempted to use PSLQ to figure out whether $\zeta(3)/\pi^3$ was a low-degree low-height algebraic number. Result: If it has degree $\le10$ then its height is at least $10^{91}$."

This was posted by someone identifying himself as Warren Smith.

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I can't find references but I know it has been shown that if $\zeta(3)/\pi^3=a/b$ is rational then $a$ and $b$ are enormous.