  A folklore conjecture in number theory states that the only integers whose
expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and
$82000$. In this paper, we present the first progress on this conjecture.
Furthermore, we investigate the density of the integers containing only binary
digits in their base $3$ or $4$ expansion, whereon an exciting transition in
behaviour is observed. Our methods shed light on the reasons for this, and
relate to several well-known questions, such as Graham's problem and a related
conjecture of Pomerance. Finally, we generalise this setting and prove that the
set of numbers in $[0, 1]$ who do not contain some digit in their $b$-expansion
for all $b \geq 3$ has zero Hausdorff dimension.
