Let $k \ne 0$ and $3\leq d_1 < d_2 < \cdots < d_r$ be integers of gcd equal to $1$ such that
$$\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$$
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 $a_i$ is divisible by $d_i ^ k$ and has only the digits $0, 1$ when
written in base $d_i$?
Conjectured by Burr, Erdős, Graham, and Li [BEGL96]
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lemma erdos124.ne_zero : answer(sorry) ↔
∀ k ≠ 0, ∀ D : Finset ℕ, (∀ d ∈ D, 3 ≤ d) → 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ → D.gcd id = 1 →
∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d k
