For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$.
Has this conjecture been proved, refuted or neither?
Is there an estimate of the minimum length of $q_b$ containing all the digits?
For example, I would expect something like $2b$ or $b^2$ for any given $q_b$.
If not refuted:
I suppose that it is not true for transcendental numbers. Is that correct?
How can we construct a transcendental number $q_b$ which does not contain all the digits?