Problem 124

Erdős Problem #124: declared status 'open'. Formalized: yes. For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq d_1<d_2<\cdots <d_r$ be integers such that$$\sum_{1\leq i\leq r}\frac{1}{d_r-1}\geq 1.$$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 $a_i\in P(d_i,0)$? If we further have $\mathrm{gcd}(d_1,\ldots,d_r)=1$ then, for any $k\geq 1$, 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 $a_i\in P(d_i,k)$? Current best: In [BEGL96] they record that Pomerance observed that the condition $\sum 1/(d_i-1)\geq 1$ is necessary (for both questions), but give no details. Prize: no. Tags: base representations, number theory.