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Wikipedia says that

A real number, having a different base-$b$ expansion for each integer $b ≥ 2$, may be normal in one base but not in another. For bases $r$ and $s$ with $\log r\ / \log s$ rational (so that $r = b^m$ and $s = b^n$) every number normal in base $r$ is normal in base $s$. For bases $r$ and $s$ with $\log r\ / \log s$ irrational, there are uncountably many numbers normal in each base but not the other.

Is there a simple example of such a number? My intuition for a possible simple example would be that a Champernowne constant $C_b$, which is known to be normal is base $b$, might not be normal in some other base, but apparently this is an open question.

(P.S. Does this count as a "number theory" question or an "elementary number theory" question?)

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There are two closely related example to my knowledge.

For one, the number $$ \alpha_{b,c} = \sum_{n=1}^\infty \frac{1}{c^n b^{c^n}} $$ is normal base-$b$ provided $b$ and $c$ are coprime integers $>1$. However, it is known that this number is not normal to several different bases, including base $bc$. Further details can be found in http://link.springer.com/article/10.1007/s11139-012-9417-3.

Wagner studied a similar construction and was able to form whole rings all of whose non-zero elements were normal to one base but not normal to another. A google search for Wagner rings will toss up a few papers on this subject.

So far as I am aware, the problem is still open for Champernowne's constant in any base.

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