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Erdős problem 283

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Let p ⁣:Z→Zp\colon \mathbb{Z} \rightarrow \mathbb{Z} be a polynomial whose leading coefficient is positive and such that there exists no d≥2d≥2 with d∣p(n)d ∣ p(n) for all n≥1n≥1. Is it true that, for all sufficiently large mm, there exist integers 1≤n1<⋯<nk1≤n_1<\dots < n_k such that 1=1n1+⋯+1nk1=\frac{1}{n_1}+\cdots+\frac{1}{n_k} and m=p(n1)+⋯+p(nk)m=p(n_1)+\cdots+p(n_k)?

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8 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

283.lean

Retained formal statement3 of 8

Burr has proved this if p(x)=xkp(x)=x^k with k≥1k\geq 1 and if we allow ni=njn_i=n_j.

FormalConjectures/ErdosProblems/283.leanErdos283.erdos_283.variants.burr12 linesExact file
∀ k ≥ 1,  ∀ᶠ (m : ℕ) in Filter.atTop,    ∃ M,      (∀ n ∈ M, 1 ≤ n) ∧        (Multiset.map (fun n => 1 / n) do                let a ← M                pure ↑a).sum =            1 ∧          (Multiset.map (fun n => n ^ k) do                let a ← M                pure ↑a).sum =            ↑m
SolvedStatement only, no proof