24356.lean`

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6 declarations, 1 contain sorry, 0 declared as axioms

Determinant of Hankel matrix of the first $2n-1$ prime numbers

The determinant of the $n \times n$ Hankel matrix whose entries are the first $2n-1$ prime numbers. The matrix $M$ has entries $M_{i, j} = p_{i+j}$ for $i, j \in \{0, \dots, n-1\}$, where $p_k = \mathrm{Nat.nth\;Nat.Prime} (k)$ is the $k$-th prime starting at $p_0=2$. $a(0)=1$ by convention.

References:

A024356

Imports (1)

FormalConjecturesUtil

def OeisA24356.a The determinant of the $n \times n$ Hankel matrix of primes line 36 source
Source documentation
The determinant of the $n \times n$ Hankel matrix of primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Matrix.det, Fin, Equiv, Matrix, Matrix.of, Nat.nth, Nat.Prime, HAdd.hAdd
11 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℤ :=
…
```
theorem OeisA24356.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, Int, OeisA24356.a, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Matrix.det_fin_zero, Equiv, Fin, Matrix, Matrix.of, Nat.nth, Nat.Prime, HAdd.hAdd, eq_self
18 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
…
```
Mentions: OeisA24356.a
theorem OeisA24356.a_1 Value of the sequence a at 1 line 46 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA24356.a, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Matrix.det, Fin, Equiv, Matrix, Matrix.of, Nat.nth, Nat.Prime, HAdd.hAdd, Inhabited.default, Matrix.det.congr_simp, funext, congr, Fin.val_eq_zero, add_zero, Nat.nth_prime_zero_eq_two and 2 more
22 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 = 2 := by
…
```
Mentions: OeisA24356.a
theorem OeisA24356.a_2 Value of the sequence a at 2 line 51 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA24356.a, OfNat.ofNat, Nat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Matrix.det, Fin, Equiv, Matrix, Matrix.of, Nat.nth, Nat.Prime, HAdd.hAdd, HSub.hSub, HMul.hMul, Matrix.det_fin_two, congr, add_zero, Nat.nth_prime_zero_eq_two, Nat.mod_succ and 4 more
32 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: OeisA24356.a
theorem OeisA24356.a_4 Value of the sequence a at 4 line 57 source
Source documentation
Value of the sequence a at 4. This records the zero whose uniqueness is conjectured below.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA24356.a, OfNat.ofNat, Nat
Proof uses: id, Nat.Prime, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth, Nat.nth_count, Matrix, Fin, Equiv, Matrix.of, HAdd.hAdd, Matrix.vecCons, Matrix.vecEmpty, Matrix.ext, List, List.Mem, List.finRange and 62 more
63 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 0 := by
…
```
Mentions: OeisA24356.a
theorem OeisA24356.conjecture "I conjecture that $a(4)$ is the only zero sorry line 85 source
Source documentation
"I conjecture that $a(4)$ is the only zero. - Jon Perry, Mar 22 2004"
Stated as a biconditional: the claim that $a(4)$ is the only zero asserts both that $a(4) = 0$ and that no other index vanishes. A bare implication a n = 0 → n = 4 would be satisfied vacuously by a sequence with no zero at all. The existence direction is certified by a_4; the uniqueness direction is the open part.

depends on sorry what this means
Statement uses: Nat, Iff, Eq, Int, OeisA24356.a, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture : ∀ n : ℕ, a n = 0 ↔ n = 4 := by
…
```
Mentions: OeisA24356.a