FormalConjectures.ErdosProblems.«1210»

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import FormalConjecturesUtil

Erdős Problem 1210

References:

erdosproblems.com/1210

[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.

[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.


open Finset


namespace Erdos1210

Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$. Is it true that $\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p < n}\frac{1}{p}+O(1)$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1210 :
  answer(sorry) ↔
    ∃ C : ℝ, ∀ n : ℕ, ∀ A : Finset ℕ,
      (∀ a ∈ A, 1 ≤ a ∧ a < n) →
      (∀ a ∈ A, ∀ b ∈ A, a ≠ b → a.Coprime b) →
      ∑ a ∈ A, (1 / ((n : ℝ) - a)) ≤ (∑ p ∈ (range n).filter Prime, (1 / (p : ℝ))) + C := by⊢ True ↔
  ∃ C,
    ∀ (n : ℕ) (A : Finset ℕ),
      (∀ a ∈ A, 1 ≤ a ∧ a < n) →
        (∀ a ∈ A, ∀ b ∈ A, a ≠ b → a.Coprime b) → ∑ a ∈ A, 1 / (↑n - ↑a) ≤ ∑ p ∈ range n with Prime p, 1 / ↑p + C
  sorryAll goals completed! 🐙

In [Er80] he claims he "did not state this quite correctly" in [Er77c]. The problem in [Er77c] which Erdős is presumably referring to states that if $n < q_1 < \cdots < q_k\leq m$ is the set of primes in $(n,m]$ then $\sum \frac{1}{q_i-n} < \sum_{p < m-n}\frac{1}{p}+O(1)$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1210.variants.er80_correction :
  answer(sorry) ↔
    ∃ C : ℝ, ∀ n m : ℕ, n < m →
      ∑ q ∈ (Ioc n m).filter Prime, (1 / ((q : ℝ) - n)) <
      (∑ p ∈ (range (m - n)).filter Prime, (1 / (p : ℝ))) + C := by⊢ True ↔
  ∃ C, ∀ (n m : ℕ), n < m → ∑ q ∈ Ioc n m with Prime q, 1 / (↑q - ↑n) < ∑ p ∈ range (m - n) with Prime p, 1 / ↑p + C
  sorryAll goals completed! 🐙


end Erdos1210