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May 17, 2015 at 2:06 comment added Noam D. Elkies You might get to $10^{-9}$ this way, but never to $10^{-900}$. The approach via simultaneous Diophantine approximation (using the LLL algorithm, or better yet PSLQ) is polynomial in the exponent and readily solve the question for $10^{-900}$ (as was done by O.S.Dawg) and beyond.
May 13, 2015 at 1:02 history answered Robert Israel CC BY-SA 3.0