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

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 $q_b$.

3. 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?

Thanks