71524.lean`

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

9 declarations, 2 contain sorry, 0 declared as axioms

Determinant of matrix with entries indicating primality of $i^2 + j^2$

The sequence $a(n)$ is the determinant of the $n \times n$ matrix $M$ defined by $M(i,j) = 1$ if $i^2 + j^2$ is prime, and $0$ otherwise, where $1 \le i, j \le n$.

References:

A071524

Imports (1)

FormalConjecturesUtil

def OeisA71524.a Determinant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^2 + (j+1)^2$ is prime, and $0$ otherwise line 38 source
Source documentation
Determinant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^2 + (j+1)^2$ is prime, and $0$ otherwise.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Matrix, Fin, HAdd.hAdd, OfNat.ofNat, ite, Nat.Prime, HPow.hPow, Matrix.det
13 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℤ :=
…
```
theorem OeisA71524.a_1 Value of the sequence a at 1 line 47 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71524.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
5 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
…
```
Mentions: OeisA71524.a
theorem OeisA71524.a_2 Value of the sequence a at 2 line 52 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71524.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
7 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 = -1 := by
…
```
Mentions: OeisA71524.a
theorem OeisA71524.a_3 Value of the sequence a at 3 line 57 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71524.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
7 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 = -1 := by
…
```
Mentions: OeisA71524.a
theorem OeisA71524.a_4 Value of the sequence a at 4 line 62 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71524.a, OfNat.ofNat, Nat
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 = 1 := by
…
```
Mentions: OeisA71524.a
theorem OeisA71524.a_5 Value of the sequence a at 5 line 67 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA71524.a, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 1 := by
…
```
Mentions: OeisA71524.a
def OeisA71524.generalDet Determinant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^{2^m} + (j+1)^{2^m}$ is prime, and $0$ otherwise line 73 source
Source documentation
Determinant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^{2^m} + (j+1)^{2^m}$ is prime, and $0$ otherwise.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Matrix, Fin, HAdd.hAdd, OfNat.ofNat, ite, Nat.Prime, HPow.hPow, Matrix.det
13 instances, coercions and auxiliary constants left out.
```
def generalDet (m n : ℕ) : ℤ :=
…
```
theorem OeisA71524.conjecture1 Conjecture: $a(n) = 0$ for no $n > 28$ sorry line 85 source
Source documentation
Conjecture: $a(n) = 0$ for no $n > 28$.
- Zhi-Wei Sun, Aug 26 2013
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Ne, Int, OeisA71524.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 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 : 28 < n) : a n ≠ 0 := by
…
```
Mentions: OeisA71524.a
theorem OeisA71524.conjecture2 Conjecture (Generalization): For every $m \in \mathbb{N}$, the determinant generalDet m n is nonzero for all sufficiently large $n$ sorry line 94 source
Source documentation
Conjecture (Generalization): For every $m \in \mathbb{N}$, the determinant generalDet m n is nonzero for all sufficiently large $n$.
- Zhi-Wei Sun, Aug 26-27 2013
depends on sorry what this means
Statement uses: Nat, Exists, LT.lt, Ne, Int, OeisA71524.generalDet, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (m : ℕ) : ∃ N : ℕ, ∀ n : ℕ, N < n → generalDet m n ≠ 0 := by
…
```
Mentions: OeisA71524.generalDet