All Random Solved Random Open
Let $3\leq d_1<d_2<\cdots <d_k$ be integers such that \[\sum_{1\leq i\leq k}\frac{1}{d_i-1}\geq 1.\] Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i\in \{0,1\}$ and $a_i$ has only the digits $0,1$ when written in base $d_i$?
Conjectured by Burr, Erdős, Graham, and Li [BEGL96]. Pomerance observed that the condition $\sum 1/(d_i-1)\geq 1$ is necessary. In [BEGL96] they prove the property holds for $\{3,4,7\}$.

See also [125].

Additional thanks to: Boris Alexeev and Dustin Mixon