Erdős problem 124 · problems.science

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 $d_i$ ?

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124.lean

Retained formal statement5 of 5

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.

FormalConjectures/ErdosProblems/124.leanErdos124.erdos124.zero4 linesExact file

1True ↔2  ∀ (D : Finset ℕ),3    (∀ d ∈ D, 3 ≤ d) →4      1 ≤ ∑ d ∈ D, (↑d - 1)⁻¹ → ∀ᶠ (n : ℕ) in Filter.atTop, n ∈ ∑ d ∈ D, Erdos124.sumsOfDistinctPowers d 0

SolvedStatement only, no proof

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