FormalConjectures/ErdosProblems/770.lean

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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

7 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 770

References:

Imports (1)
Imported by 1 file
  1. def Erdos770.h Let $h n$ be the minimal number such that $2 ^ n - 1, \dots, h(n) ^ n - 1$ are collectively coprime line 40 source

    Source documentation

    Let $h n$ be the minimal number such that $2 ^ n - 1, \dots, h(n) ^ n - 1$ are collectively coprime.

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

    Statement uses: Nat, ENat

    Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, OfNat.ofNat, Eq, Finset.gcd, Finset.image, HSub.hSub, HPow.hPow, Finset.Icc, ENat.toNat, id

    24 instances, coercions and auxiliary constants left out.

    noncomputable def h (n : ℕ) : ℕ∞ := sInf {m | 2 < m ∧
…
  2. theorem Erdos770.Nat.Prime.h_eq_add_one n + 1 is prime iff h n = n + 1 sorry line 45 source

    Source documentation

    n + 1 is prime iff h n = n + 1. This is described as 'easy to see' in [Er74b].

    depends on sorry what this means

    Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Eq, ENat, Erdos770.h, HAdd.hAdd, Nat.Prime

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

    20 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem Nat.Prime.h_eq_add_one {n : ℕ} (hn : 2 < n) : h n = n + 1 ↔ (n + 1).Prime := by
…

    Mentions: Erdos770.h

  3. theorem Erdos770.erdos_770.variants.odd_h_unbounded For odd n, the values of h n form an unbounded set sorry line 51 source

    Source documentation

    For odd n, the values of h n form an unbounded set. This is described as 'easy to see' in [Er74b].

    depends on sorry what this means

    Statement uses: Set.Unbounded, Nat, LE.le, Set.image, ENat, ENat.toNat, Erdos770.h, Odd

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

    5 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_770.variants.odd_h_unbounded : Unbounded (· ≤ ·) (ENat.toNat '' (h '' Odd)):= by
…

    Mentions: Erdos770.h

  4. theorem Erdos770.erdos_770.parts.i For every prime p, does the density of integers with h n = p exist? sorry line 57 source

    Source documentation

    For every prime p, does the density of integers with h n = p exist?

    depends on sorry what this means

    Statement uses: Iff, True, Nat, Nat.Prime, Exists, Real, Set.HasDensity, Set.ofPred, Eq, ENat, Erdos770.h, Set.univ

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

    13 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_770.parts.i : answer(sorry) ↔ ∀ p : ℕ, p.Prime → ∃ a, HasDensity {n | h n = p} a := by
…

    Mentions: Erdos770.h

  5. theorem Erdos770.erdos_770.parts.ii Does liminf h n = ∞? sorry line 62 source

    Source documentation

    Does liminf h n = ∞?

    depends on sorry what this means

    Statement uses: Iff, True, Eq, ENat, Filter.liminf, Nat, Erdos770.h, Filter.atTop, Top.top

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

    13 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_770.parts.ii : answer(sorry) ↔ liminf h atTop = ⊤ := by
…

    Mentions: Erdos770.h

  6. theorem Erdos770.erdos_770.parts.iii Is it true that if p is the greatest prime such that p - 1 ∣ n and p > n ^ ε, then h n = p? sorry line 68 source

    Source documentation

    Is it true that if p is the greatest prime such that p - 1 ∣ n and p > n ^ ε, then h n = p?

    depends on sorry what this means

    Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, SupSet.sSup, Set.ofPred, And, Nat.Prime, Dvd.dvd, HSub.hSub, HPow.hPow, Eq, ENat, Erdos770.h, Filter.atTop

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

    28 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_770.parts.iii : answer(sorry) ↔ ∀ ε > 0, ∀ᶠ n in atTop,
    let p := sSup {m : ℕ | m.Prime ∧ m - 1 ∣ n}
…
  7. theorem Erdos770.erdos_770.variants.three It is probably true that h n = 3 for infinitely many n sorry line 75 source

    Source documentation

    It is probably true that h n = 3 for infinitely many n.

    depends on sorry what this means

    Statement uses: Set.Infinite, Nat, Set.ofPred, Eq, ENat, Erdos770.h, 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_770.variants.three : {n | h n = 3}.Infinite := by
…

    Mentions: Erdos770.h