FormalConjectures.ErdosProblems.«1073»

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import FormalConjecturesUtil

Erdős Problem 1073

Reference: erdosproblems.com/1073


open Nat Filter


namespace Erdos1073

Let $A(x)$ count the number of composite $u < x$ such that $n!+1 \equiv 0 (\mod u)$ for some $n$.

noncomputable def A (x : ℕ) : ℝ := {u | u.Composite ∧ ∃ n, n ! + 1 ≡ 0 [MOD u] ∧ u < x}.ncard

Is it true that $A(x) \le x^{o(1)}$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1073 :
    answer(sorry) ↔ ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧ ∀ x, A x ≤ x ^ (o x) := by⊢ True ↔ ∃ o, o =o[atTop] 1 ∧ ∀ (x : ℕ), A x ≤ ↑x ^ o x
  sorryAll goals completed! 🐙


end Erdos1073