367.lean`

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L4DEL: Lean4 Domain Explanation Language

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

6 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 367

References:

erdosproblems.com/367
[ErGr80] P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, L'Enseignement Mathématique (1980).

Imports (1)

FormalConjecturesUtil

def Erdos367.B B r n is the $r$-full part of $n$: the product of prime powers $p^a \| n$ with $a \geq r$ line 37 source
Source documentation
B r n is the $r$-full part of $n$: the product of prime powers $p^a \| n$ with $a \geq r$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.prod, Finset.filter, LE.le, Finsupp, Nat.factorization, Finsupp.support, HPow.hPow
11 instances, coercions and auxiliary constants left out.
```
def B (r n : ℕ) : ℕ :=
…
```
theorem Erdos367.erdos_367.parts.i Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once) sorry line 46 source
Source documentation
Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k \geq 1$, $\prod_{n \leq m < n+k} B_2(m) \ll n^{2+o(1)}$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Real, And, Asymptotics.IsLittleO, Filter.atTop, Filter.Eventually, Finset.prod, Finset.Ico, HAdd.hAdd, Erdos367.B, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_367.parts.i : answer(sorry) ↔ ∀ k : ℕ, 1 ≤ k →
    ∃ e : ℕ → ℝ,
      e =o[atTop] (1 : ℕ → ℝ) ∧
      ∀ᶠ n in atTop,
        ((∏ m ∈ .Ico n (n + k), B 2 m : ℕ) : ℝ) ≤ (n : ℝ) ^ (2 + e n) := by
…
```
Mentions: Erdos367.B
theorem Erdos367.erdos_367.parts.ii Or perhaps even $\prod_{n \leq m < n+k} B_2(m) \ll_k n^2$? sorry line 59 source
Source documentation
Or perhaps even $\prod_{n \leq m < n+k} B_2(m) \ll_k n^2$?
van Doorn notes in the comments that this fails for all $k \geq 3$.

depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Asymptotics.IsBigO, Real, Filter.atTop, Finset.prod, Finset.Ico, HAdd.hAdd, Erdos367.B, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_367.parts.ii : answer(False) ↔ ∀ k : ℕ, 1 ≤ k →
    (fun n ↦ ((∏ m ∈ .Ico n (n + k), B 2 m : ℕ) : ℝ)) =O[atTop]
      fun n ↦ (n : ℝ) ^ (2 : ℝ) := by
…
```
Mentions: Erdos367.B
theorem Erdos367.erdos_367.variants.k_le_two van Doorn notes in the comments that for $k \leq 2$ we trivially have $\prod_{n \leq m < n+k} B_2(m) \ll n^2$ sorry line 69 source
Source documentation
van Doorn notes in the comments that for $k \leq 2$ we trivially have $\prod_{n \leq m < n+k} B_2(m) \ll n^2$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Asymptotics.IsBigO, Real, Filter.atTop, Finset.prod, Finset.Ico, HAdd.hAdd, Erdos367.B, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_367.variants.k_le_two : ∀ k : ℕ, k ≤ 2 →
    (fun n ↦ ((∏ m ∈ .Ico n (n + k), B 2 m : ℕ) : ℝ)) =O[atTop]
      fun n ↦ (n : ℝ) ^ (2 : ℝ) := by
…
```
Mentions: Erdos367.B
theorem Erdos367.erdos_367.variants.k_ge_three_lower But this fails for all $k \geq 3$, and in fact $\prod_{n \leq m < n+3} B_2(m) \gg n^2 \log n$ infinitely often sorry line 79 source
Source documentation
But this fails for all $k \geq 3$, and in fact $\prod_{n \leq m < n+3} B_2(m) \gg n^2 \log n$ infinitely often.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Frequently, Nat, LE.le, HMul.hMul, HPow.hPow, Real.log, Finset.prod, Finset.Ico, HAdd.hAdd, Erdos367.B, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_367.variants.k_ge_three_lower :
    ∃ c > (0 : ℝ), ∃ᶠ (n : ℕ) in atTop,
      c * ((n : ℝ) ^ 2 * Real.log (n : ℝ)) ≤
        ((∏ m ∈ .Ico n (n + 3), B 2 m : ℕ) : ℝ) := by
…
```
Mentions: c, Erdos367.B
theorem Erdos367.erdos_367.variants.higher_full_parts Documented declaration: higher_full_parts sorry line 98 source
Source documentation
It would also be interesting to find upper and lower bounds for the analogous product with $B_r$ for $r \geq 3$, where $B_r(n)$ is the $r$-full part of $n$ (that is, the product of prime powers $p^a \mid n$ such that $p^{a+1} \nmid n$ and $a \geq r$). Is it true that, for every fixed $r \geq 3$ and $k \geq 2$, there is some $\epsilon = \epsilon(r, k) > 0$ with $\limsup \frac{\prod_{n \leq m < n+k} B_r(m)}{n^{1+\epsilon}} = \infty$?
The webpage writes "for every fixed $r,k \geq 2$ and $\epsilon > 0$", but van Doorn observes in the comments that the intended reading lets $\epsilon$ depend on $r$ and $k$. Since $B_r(m) \mid m$, the product is at most $\prod_{n \leq m < n+k} m$, so the reading with $\epsilon$ universally quantified is false.

L4DEL explanation
It would also be interesting to find upper and lower bounds for the analogous product with $B_r$ for $r \geq 3$, where $B_r(n)$ is the $r$-full part of $n$ (that is, the product of prime powers $p^a \mid n$ such that $p^{a+1} \nmid n$ and $a \geq r$). Is it true that, for every fixed $r \geq 3$ and $k \geq 2$, there is some $\epsilon = \epsilon(r, k) > 0$ with $\limsup \frac{\prod_{n \leq m < n+k} B_r(m)}{n^{1+\epsilon}} = \infty$? The webpage writes "for every fixed $r,k \geq 2$ and $\epsilon > 0$", but van Doorn observes in the comments that the intended reading lets $\epsilon$ depend on $r$ and $k$. Since $B_r(m) \mid m$, the product is at most $\prod_{n \leq m < n+k} m$, so the reading with $\epsilon$ universally quantified is false.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Real, And, LT.lt, Eq, EReal, Filter.limsup, HDiv.hDiv, Finset.prod, Finset.Ico, HAdd.hAdd, Erdos367.B, HPow.hPow, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_367.variants.higher_full_parts : answer(sorry) ↔
    ∀ r k : ℕ, 3 ≤ r → 2 ≤ k → ∃ ε : ℝ, 0 < ε ∧
      atTop.limsup (fun n ↦
        ((∏ m ∈ .Ico n (n + k), B r m : ℕ) / (n : ℝ) ^ (1 + ε) |>.toEReal)) = ⊤ := by
…
```
Mentions: Erdos367.B