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Erdős problem 1190

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Let ϵm=max⁡∑1ni\epsilon_m=\max \sum \frac{1}{n_i} where the maximum is taken over all finite sequences m<n1<⋯<nkm<n_1<\cdots<n_k for which there exist congruences ai(modni)a_i\pmod{n_i} such that no integer satisfies two such congruences.

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5 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

1190.lean

Retained formal statement3 of 5

Erdős [Er80] seems to credit Mirsky and Newman with the result that ϵm<1\epsilon_m<1, but gives no reference.

FormalConjectures/ErdosProblems/1190.leanErdos1190.erdos_1190.variants.lt_one5 linesExact file
∀ (m : ℕ),  1 ≤ m →    ∀ (S : Finset ℕ) (a : ℕ → ℤ),      (∀ n ∈ S, m < n) →        (∀ n ∈ S, ∀ n' ∈ S, n ≠ n' → ¬∃ x, x ≡ a n [ZMOD ↑n] ∧ x ≡ a n' [ZMOD ↑n']) → ∑ n ∈ S, (↑n)⁻¹ < 1
SolvedStatement only, no proof