727.lean`

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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 727

Reference: erdosproblems.com/727

Imports (1)

FormalConjecturesUtil

theorem Erdos727.erdos_727 Let $k ≥ 2$ sorry line 36 source
Source documentation
Let $k ≥ 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many $n$?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Set.Infinite, Set.ofPred, Dvd.dvd, HPow.hPow, Nat.factorial, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_727 : answer(sorry) ↔ ∀ k ≥ 2,
    Set.Infinite {n : ℕ | (Nat.factorial (n + k)) ^ 2 ∣ Nat.factorial (2 * n)} := by
…
```
theorem Erdos727.erdos_727.variants.k_2 It is open even for $k = 2$ sorry line 45 source
Source documentation
It is open even for $k = 2$. Let $k = 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many n?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Dvd.dvd, HPow.hPow, Nat.factorial, HAdd.hAdd, OfNat.ofNat, HMul.hMul
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 erdos_727.variants.k_2 :
    letI k := 2
…
```
theorem Erdos727.erdos_727.variants.k_1 Balakran proved this holds for $k = 1$ sorry line 56 source
Source documentation
Balakran proved this holds for $k = 1$.
Let $k = 1$. Does $((n+k)!)^2∣(2n)!$ for infinitely many $n$?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Dvd.dvd, HPow.hPow, Nat.factorial, HAdd.hAdd, OfNat.ofNat, HMul.hMul
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 erdos_727.variants.k_1 :
    letI k := 1
…
```
theorem Erdos727.erdos_727.variants.k_1_2 Erdős, Graham, Ruzsa, and Straus observe that the method of Balakran can be further used to prove that there are infinitely many $n$ such that $(n+k)!(n+1)!∣(2n)!$ sorry line 66 source
Source documentation
Erdős, Graham, Ruzsa, and Straus observe that the method of Balakran can be further used to prove that there are infinitely many $n$ such that $(n+k)!(n+1)!∣(2n)!$

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Set.Infinite, Set.ofPred, Dvd.dvd, HMul.hMul, Nat.factorial, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_727.variants.k_1_2 (k : ℕ) (hk : 2 ≤ k) :
    Set.Infinite {n : ℕ |
      (Nat.factorial (n + k)) * (Nat.factorial (n + 1)) ∣ Nat.factorial (2 * n)} := by
…
```