1818.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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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

9 declarations, 2 contain sorry, 0 declared as axioms

Squares of double factorials

Squares of double factorials: $a(n) = ((2n-1)!!)^2 = (1 \cdot 3 \cdot 5 \cdots (2n-1))^2$.

References:

Imports (1)

FormalConjecturesUtil

def OeisA1818.a The sequence of squares of double factorials: $a(n) = ((2n-1)!!)^2$ line 36 source
Source documentation
The sequence of squares of double factorials: $a(n) = ((2n-1)!!)^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HPow.hPow, Finset.prod, Finset.range, HAdd.hAdd, HMul.hMul, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA1818.a_0 Value of the sequence a at 0 line 41 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1818.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by decide
…
```
Mentions: OeisA1818.a
theorem OeisA1818.a_1 Value of the sequence a at 1 line 45 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1818.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by decide
…
```
Mentions: OeisA1818.a
theorem OeisA1818.a_2 Value of the sequence a at 2 line 49 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1818.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 9 := by decide
…
```
Mentions: OeisA1818.a
theorem OeisA1818.a_3 Value of the sequence a at 3 line 53 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1818.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 225 := by decide
…
```
Mentions: OeisA1818.a
theorem OeisA1818.a_4 Value of the sequence a at 4 line 57 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA1818.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 11025 := by decide
```
Mentions: OeisA1818.a
def OeisA1818.fEntry Characteristic function $f(j, k)$ for matrix entries in $\mathbb{Z}/p^2\mathbb{Z}$ line 60 source
Source documentation
Characteristic function $f(j, k)$ for matrix entries in $\mathbb{Z}/p^2\mathbb{Z}$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, ZMod, HPow.hPow, OfNat.ofNat
Proof uses: ite, Eq, Int, HAdd.hAdd, HSub.hSub, HMul.hMul, Inv.inv
31 instances, coercions and auxiliary constants left out.
```
noncomputable def fEntry {p : ℕ} (i j : ℕ) : ZMod (p ^ 2) :=
…
```
theorem OeisA1818.conjecture1 Conjecture 1: For any primitive sorry line 78 source
Source documentation
Conjecture 1: For any primitive $2n$-th root $\zeta$ of unity, the permanent of the $2n \times 2n$ matrix $[m(j,k)]_{j,k=1..2n}$ coincides with $a(n) = ((2n-1)!!)^2$, where $m(j,k)$ is $(1+\zeta^{j-k})/(1-\zeta^{j-k})$ if $j \neq k$, and $1$ otherwise.
- Zhi-Wei Sun, Dec 21 2021
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Complex, IsPrimitiveRoot, HMul.hMul, Eq, Matrix.permanent, Fin, ite, HDiv.hDiv, HAdd.hAdd, HPow.hPow, Int, HSub.hSub, OeisA1818.a
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 conjecture1 (n : ℕ) (hn : 1 ≤ n) :
    ∀ (ζ : ℂ), IsPrimitiveRoot ζ (2 * n) →
      Matrix.permanent (fun (i j : Fin (2 * n)) =>
        if i = j then
          (1 : ℂ)
        else
          (1 + ζ ^ (i.val - j.val : ℤ)) / (1 - ζ ^ (i.val - j.val : ℤ))
      ) = (a n : ℂ) := by
…
```
Mentions: OeisA1818.a
theorem OeisA1818.conjecture2 Conjecture 2: Let $p$ be an odd prime sorry line 95 source
Source documentation
Conjecture 2: Let $p$ be an odd prime. Then the permanent of the $(p-1) \times (p-1)$ matrix $[f(j,k)]_{j,k=1..p-1}$ is congruent to $a((p-1)/2) = ((p-2)!!)^2 \pmod{p^2}$, where $f(j,k)$ is $(j+k)/(j-k)$ if $j \neq k$, and $f(j,k) = 1$ otherwise.
- Zhi-Wei Sun, Dec 22 2021
depends on sorry what this means
Statement uses: Nat, Nat.Prime, Ne, OfNat.ofNat, HSub.hSub, ZMod, HPow.hPow, Fin, Matrix, OeisA1818.fEntry, HAdd.hAdd, Eq, Matrix.permanent, OeisA1818.a, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 {p : ℕ} (hp : p.Prime) (h_odd : p ≠ 2) :
    let N : ℕ := p - 1
…
```