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 $b\ge 3$ has zero Hausdorff dimension.

数论中的一个民俗猜想指出，只有整数在基数中展开的整数 $3，4$ 和5仅包含二进制数字是 $0，\mathrm{1个}$和82000。在本文中，我们介绍了这一猜想的第一个进展。此外，我们研究了以3或4为底扩展的仅包含二进制数字的整数的密度，从而观察到了行为上的激动人心的过渡。我们的方法阐明了造成这种情况的原因，并且涉及到几个众所周知的问题，例如Graham问题和相关的Pomerance猜想。最后，我们对此设置进行概括，并证明$[0，\mathrm{1个}]$谁的b扩展中都不含数字$b\ge 3$ Hausdorff维数为零。