FormalConjectures/ErdosProblems/1073.lean

GitHub source file

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1073

Reference: erdosproblems.com/1073

Imports (1)
  1. def Erdos1073.A Let $A(x)$ count the number of composite $u < x$ such that $n!+1 \equiv 0 (\mod u)$ for some $n$ line 35 source

    Source documentation

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

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat, Real

    Proof uses: Set.ncard, Set.ofPred, And, Nat.Composite, Exists, Nat.ModEq, HAdd.hAdd, Nat.factorial, OfNat.ofNat, LT.lt

    6 instances, coercions and auxiliary constants left out.

    noncomputable def A (x : ℕ) : ℝ := {u | u.Composite ∧ ∃ n, n ! + 1 ≡ 0 [MOD u] ∧ u < x}.ncard
…
  2. theorem Erdos1073.erdos_1073 Is it true that $A(x) \le x^{o(1)}$? sorry line 39 source

    Source documentation

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

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, LE.le, Erdos1073.A, HPow.hPow

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    21 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_1073 :
    answer(sorry) ↔ ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧ ∀ x, A x ≤ x ^ (o x) := by
…

    Mentions: Erdos1073.A