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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. – Chris Eagle Dec 13 '12 at 18:58
Out of pure interest, does "normal" mean that every digit appears equally "often"? – CBenni Dec 13 '12 at 19:26
I think for this special case, calculate the probability that a real number has decimal expression with only even digits might be helpful. – ougao Dec 13 '12 at 19:27
normal number – ougao Dec 13 '12 at 19:28
@ougao: How will that help? – Chris Eagle Dec 13 '12 at 19:35

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 :) – Mahan Dec 14 '12 at 6:34
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. – Gerry Myerson Dec 14 '12 at 11:58

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