TITLE: Square root of an integer has only even digits
QUESTION [12 upvotes]: Is there a non-square positive integer $n$, that $\sqrt{n}$ has only even digits in its decimal representation ?

REPLY [3 votes]: 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.