FormalConjectures/ErdosProblems/890.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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 890

Reference:

Imports (1)
Imported by 1 file
  1. def Erdos890.omegaGt omegaGt k n counts the number of distinct prime factors of n that are strictly greater than k line 38 source

    Source documentation

    omegaGt k n counts the number of distinct prime factors of n that are strictly greater than k.

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

    Statement uses: Nat

    Proof uses: Finset.card, Finset.filter, GT.gt, Nat.primeFactors

    2 instances, coercions and auxiliary constants left out.

    def omegaGt (k n : ℕ) : ℕ :=
…
  2. theorem Erdos890.erdos_890.parts.a Documented declaration: a sorry line 49 source

    Source documentation

    If $\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$, then is it true that, for every $k\geq 1$, $$\liminf_{n\to \infty}\sum_{0\leq i < k}\omega_k(n+i)\leq k?$$

    depends on sorry what this means

    Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, LE.le, EReal, Filter.liminf, Finset.sum, Finset.range, Erdos890.omegaGt, HAdd.hAdd, Filter.atTop

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

    34 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_890.parts.a :
    answer(sorry) ↔
    ∀ k ≥ 1, liminf (fun n ↦ (∑ i ∈ range k, (ω_gt k (n + i) : EReal))) atTop ≤ k := by
…
  3. theorem Erdos890.erdos_890.parts.b Documented declaration: b sorry line 60 source

    Source documentation

    Is it true that $$\limsup_{n\to \infty}\left(\sum_{0\leq i < k}\omega(n+i)\right) \frac{\log\log n}{\log n}=1,$$ where $\omega$ counts the number of distinct prime factors without restriction?

    depends on sorry what this means

    Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Eq, EReal, Filter.limsup, HMul.hMul, Finset.sum, Finset.range, ArithmeticFunction, ArithmeticFunction.cardDistinctFactors, HAdd.hAdd, HDiv.hDiv, Real.log, Real, Filter.atTop

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

    54 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_890.parts.b :
    answer(sorry) ↔ ∀ k ≥ 1, limsup (fun n ↦ (∑ i ∈ range k, (ω (n + i) : EReal)) *
      (log (log n) / log n)) atTop = 1 := by
…
  4. theorem Erdos890.erdos_890.variants.liminf_lower_bound A question of Erdős and Selfridge [ErSe67], who observe that $\liminf_{n\to \infty}\sum_{0\leq i < k}\omega(n+i)\geq k+\pi(k)-1$ for every $k$ sorry line 73 source

    Source documentation

    A question of Erdős and Selfridge [ErSe67], who observe that $\liminf_{n\to \infty}\sum_{0\leq i < k}\omega(n+i)\geq k+\pi(k)-1$ for every $k$. This follows from Pólya's theorem that the set of $k$-smooth integers has unbounded gaps - indeed, $n(n+1)\cdots (n+k-1)$ is divisible by all primes $\leq k$ and, provided $n$ is large, all but at most one of $n,n+1,\ldots,n+k-1$ has a prime factor $>k$ by Pólya's theorem.

    depends on sorry what this means

    Statement uses: Nat, GE.ge, EReal, Filter.liminf, Finset.sum, Finset.range, ArithmeticFunction, ArithmeticFunction.cardDistinctFactors, HAdd.hAdd, Filter.atTop, HSub.hSub, Nat.primeCounting, OfNat.ofNat

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

    57 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_890.variants.liminf_lower_bound (k : ℕ) :
    liminf (fun n ↦ (∑ i ∈ range k, (ω (n + i) : EReal))) atTop ≥ k + π k - 1 := by
…
  5. theorem Erdos890.erdos_890.variants.omega_limsup It is a classical fact that $\limsup_{n\to \infty}\omega(n)\frac{\log\log n}{\log n}=1.$ sorry line 81 source

    Source documentation

    It is a classical fact that $\limsup_{n\to \infty}\omega(n)\frac{\log\log n}{\log n}=1.$

    depends on sorry what this means

    Statement uses: Eq, EReal, Filter.limsup, Nat, HMul.hMul, ArithmeticFunction, ArithmeticFunction.cardDistinctFactors, HDiv.hDiv, Real.log, Real, Filter.atTop, OfNat.ofNat

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

    45 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_890.variants.omega_limsup :
    limsup (fun n ↦ (ω n : EReal) * (log (log n) / log n)) atTop = 1 := by
…