FormalConjectures/ErdosProblems/459.lean

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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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 459

References:

Imports (1)
  1. def Erdos459.f The function from the problem, in its equivalent form: f u is the smallest v > u all of whose prime factors divide u line 39 source

    Source documentation

    The function from the problem, in its equivalent form: f u is the smallest v > u all of whose prime factors divide u. (Equivalently, f u is the largest v such that no m ∈ (u, v) is composed entirely of primes dividing u * v.)

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

    Statement uses: Nat

    Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, LE.le, Finset, Nat.primeFactors

    5 instances, coercions and auxiliary constants left out.

    noncomputable def f (u : ℕ) : ℕ := sInf {v | u < v ∧ v.primeFactors ⊆ u.primeFactors}
…
  2. theorem Erdos459.erdos_459 Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$ sorry line 50 source

    Source documentation

    Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$. Estimate $f(u)$.

    The estimate $u + 2 \le f(u) \le u^2$ holds for every $u \ge 2$. The upper bound is attained when $u$ is prime, and the lower bound when $u = 2^k - 2$ with $k \ge 2$; Cambie further showed that $f(n) = (1 + o(1))n$ for almost all $n$.

    depends on sorry what this means

    Statement uses: Nat, LE.le, OfNat.ofNat, And, HAdd.hAdd, Erdos459.f, HPow.hPow

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

    16 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_459 {u : ℕ} (hu : 2 ≤ u) : u + 2 ≤ f u ∧ f u ≤ u ^ 2 := by
…

    Mentions: Erdos459.f

  3. theorem Erdos459.erdos_459.variants.upper_tight The upper bound $f u ≤ u ^ 2$ is attained exactly when u is prime: $f p = p ^ 2$ sorry line 57 source

    Source documentation

    The upper bound $f u ≤ u ^ 2$ is attained exactly when u is prime: $f p = p ^ 2$.

    depends on sorry what this means

    Statement uses: Nat, Nat.Prime, Eq, Erdos459.f, HPow.hPow, OfNat.ofNat

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

    10 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_459.variants.upper_tight {p : ℕ} (hp : p.Prime) : f p = p ^ 2 := by
…

    Mentions: Erdos459.f