erdos124.zero

Let $3 \le d_1 < d_2 < \dots < d_r$ be integers 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 $a_i$ has only the digits $0, 1$ when written in base $d_i$? Conjectured by Erdős [Er97], solved by Boris Alexeev using Aristotle.

lemma erdos124.zero : answer(True) ↔
    ∀ D : Finset ℕ, (∀ d ∈ D, 3 ≤ d) → 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ →
      ∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d 0

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