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Is there a non-square positive integer $n$, that $\sqrt{n}$ has only even digits in its decimal representation ?

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    It's believed not: every irrational algebraic number is conjectured to be normal (and hence its decimal expansion contains all $10$ digits). I don't know if this special case has been solved.2012-12-13
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    Out of pure interest, does "normal" mean that every digit appears equally "often"?2012-12-13
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    I think for this special case, calculate the probability that a real number has decimal expression with only even digits might be helpful.2012-12-13
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    normal number http://en.wikipedia.org/wiki/Normal_number2012-12-13
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    @ougao: How will that help?2012-12-13

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No such $n$ is known.

If one were found, it would be the biggest shock in Mathematics since, well, maybe since ever; certainly, since Godel's incompleteness results.

No proof is known that such an $n$ does not exist.

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    Could you elaborate the connection of this question with Godel's Theorem? Thanks :)2012-12-14
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    The only connection is that Godel's results came as a shock, and finding such an $n$ would be at least as big a shock. The connection is sociological, not mathematical.2012-12-14