# Example of a number that is normal in one base but not another

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?)

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.