Editing Openai/692c9ea1-b3cc-800e-94da-4224ac78a36a (section)

=== User: I am wondering if the following problem can be attacked using Brown's criterion. ===
I am wondering if the following problem can be attacked using Brown's criterion. Could you explore this possibility?

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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, Erd\H{o}s, Graham, and Li \cite{BEGL96}. Pomerance observed that the condition $\sum 1/(d_i-1)\geq 1$ is necessary. In \cite{BEGL96} they prove the property holds for $\{3,4,7\}$.

See also [125].

References

[BEGL96] Burr, S. A. and Erd\H{o}s, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.