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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5 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

124.lean

Retained formal statement2 of 5

For any $\varepsilon > 0$ , there exists an infinite sequence $2 \le d_0 < d_1 < \dots$ such that all sufficiently large integer can be written as $\sum_{i \in I} a_i$ where $a_i$ has only the digits $0, 1$ when written in base $d_i$ , but $\sum_{i \in I} \frac 1{d_i - 1} \le \varepsilon$ .

Proved by Melfi [Me04]

FormalConjectures/ErdosProblems/124.leanErdos124.erdos124.melfi_construction6 linesExact file

1∀ {ε : ℝ},2  0 < ε →3    ∃ d,4      StrictMono d ∧5        ∑' (i : ℕ), (↑(d i) - 1)⁻¹ ≤ ε ∧6          ∀ᶠ (n : ℕ) in Filter.atTop, ∃ I a, (∀ i ∈ I, a i ∈ Erdos124.sumsOfDistinctPowers (d i) 0) ∧ ∑ i ∈ I, a i = n

SolvedStatement only, no proof

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