FormalConjectures.ErdosProblems.«688»

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import FormalConjecturesUtil

Erdős Problem 688

Reference:

erdosproblems.com/688

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


open Classical Real Filter


namespace Erdos688

Define $\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\epsilon_n} < p \leq n$ such that every integer in $[1,n]$ satisfies at least one of the congruences $\equiv a_p \pmod p$.

def Erdos688Prop (n : ℕ) (ε : ℝ) : Prop :=
  ∃ (a : ℕ → ℕ), ∀ (m : ℕ), 1 ≤ m → m ≤ n →
    ∃ (p : ℕ), p.Prime ∧ (n : ℝ)^ε < p ∧ p ≤ n ∧
      a p ≡ m [MOD p]


noncomputable def epsilonFunction (n : ℕ) : ℝ := sSup {ε : ℝ | Erdos688Prop n ε}

Estimate $\epsilon_n$ - lower bound.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_688.parts.i.lower_bound :
    (answer(sorry) : ℕ → ℝ) =O[atTop] epsilonFunction := by⊢ sorry =O[atTop] epsilonFunction
  sorryAll goals completed! 🐙

Estimate $\epsilon_n$ - upper bound.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_688.parts.i.upper_bound :
    epsilonFunction =O[atTop] (answer(sorry) : ℕ → ℝ) := by⊢ epsilonFunction =O[atTop] sorry
  sorryAll goals completed! 🐙

In particular, is it true that $\epsilon_n = o(1)$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_688.parts.ii : answer(sorry) ↔
    epsilonFunction =o[atTop] (fun (n : ℕ) ↦ (1 : ℝ)) := by⊢ True ↔ epsilonFunction =o[atTop] fun n => 1
  sorryAll goals completed! 🐙

Erdős claims in [Er80] (p. 106) that it is not difficult to prove $\epsilon_n \gg \frac{\log\log\log n}{\log\log n}$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_688.variants.lglglg_over_lglg_is_big_o :
    (fun (n : ℕ) ↦ (log (log (log (n : ℝ)))) / (log (log (n : ℝ))))
      =O[atTop] epsilonFunction := by⊢ (fun n => log (log (log ↑n)) / log (log ↑n)) =O[atTop] epsilonFunction
  sorryAll goals completed! 🐙


end Erdos688