1. FormalConjectures
  2. OEIS
  3. «24356»
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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:

@[expose] public sectionnamespace OeisA24356

The determinant of the $n \times n$ Hankel matrix of primes.

noncomputable def a (n : ℕ) : ℤ := Matrix.det (Matrix.of fun (i j : Fin n) ↦ (Nat.nth Nat.Prime (i.val + j.val) : ℤ))

Value of the sequence a at 0.

@[category test, AMS 11] theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙

Value of the sequence a at 1.

@[category test, AMS 11] theorem a_1 : a 1 = 2 := ⊢ a 1 = 2 All goals completed! 🐙

Value of the sequence a at 2.

@[category test, AMS 11] theorem a_2 : a 2 = 1 := ⊢ a 2 = 1 All goals completed! 🐙

Value of the sequence a at 4. This records the zero whose uniqueness is conjectured below.

hp13:Nat.Prime 13hp17:Nat.Prime 17h5:Nat.nth Nat.Prime 5 = 13h6:Nat.nth Nat.Prime 6 = 17hM:(Matrix.of fun i j ↦ ↑(Nat.nth Nat.Prime (↑i + ↑j))) = !![2, 3, 5, 7; 3, 5, 7, 11; 5, 7, 11, 13; 7, 11, 13, 17]⊢ !![2, 3, 5, 7; 3, 5, 7, 11; 5, 7, 11, 13; 7, 11, 13, 17].det = 0 hp13:Nat.Prime 13hp17:Nat.Prime 17h5:Nat.nth Nat.Prime 5 = 13h6:Nat.nth Nat.Prime 6 = 17hM:(Matrix.of fun i j ↦ ↑(Nat.nth Nat.Prime (↑i + ↑j))) = !![2, 3, 5, 7; 3, 5, 7, 11; 5, 7, 11, 13; 7, 11, 13, 17]⊢ !![2, 3, 5, 7; 3, 5, 7, 11; 5, 7, 11, 13; 7, 11, 13, 17].mulVec ![6, -3, -2, 1] = 0hp13:Nat.Prime 13hp17:Nat.Prime 17h5:Nat.nth Nat.Prime 5 = 13h6:Nat.nth Nat.Prime 6 = 17hM:(Matrix.of fun i j ↦ ↑(Nat.nth Nat.Prime (↑i + ↑j))) = !![2, 3, 5, 7; 3, 5, 7, 11; 5, 7, 11, 13; 7, 11, 13, 17]⊢ ![6, -3, -2, 1] 0 ∈ nonZeroDivisors ℤ hp13:Nat.Prime 13hp17:Nat.Prime 17h5:Nat.nth Nat.Prime 5 = 13h6:Nat.nth Nat.Prime 6 = 17hM:(Matrix.of fun i j ↦ ↑(Nat.nth Nat.Prime (↑i + ↑j))) = !![2, 3, 5, 7; 3, 5, 7, 11; 5, 7, 11, 13; 7, 11, 13, 17]⊢ !![2, 3, 5, 7; 3, 5, 7, 11; 5, 7, 11, 13; 7, 11, 13, 17].mulVec ![6, -3, -2, 1] = 0 All goals completed! 🐙 hp13:Nat.Prime 13hp17:Nat.Prime 17h5:Nat.nth Nat.Prime 5 = 13h6:Nat.nth Nat.Prime 6 = 17hM:(Matrix.of fun i j ↦ ↑(Nat.nth Nat.Prime (↑i + ↑j))) = !![2, 3, 5, 7; 3, 5, 7, 11; 5, 7, 11, 13; 7, 11, 13, 17]⊢ ![6, -3, -2, 1] 0 ∈ nonZeroDivisors ℤ All goals completed! 🐙

"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.

@[category research open, AMS 11 15] theorem conjecture : ∀ n : ℕ, a n = 0 ↔ n = 4 := ⊢ ∀ (n : ℕ), a n = 0 ↔ n = 4 All goals completed! 🐙end OeisA24356