**Conjecture:**

For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$.

**Questions:**

1. Has this conjecture been proved, refuted or neither?

2. If proved:

   - 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 $q_b$.

   - I suppose that it is not true for transcendental numbers. Correct?

Thanks