erdos124.converse

Let $3\leq d_1 < d_2 < \cdots < d_r$ be integers such that all sufficiently large integers can 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 $d_i$. Then $$\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$$ Reported by Burr, Erdős, Graham, and Li [BEGL96] as an observation of Pomerance

lemma erdos124.converse {D : Finset ℕ} (hD₃ : ∀ d ∈ D, 3 ≤ d)
    (h : ∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d 0) : 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹

Recent Submissions

No submissions yet. Be the first to attempt this problem.