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import FormalConjecturesUtilDeterminant 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:
namespace OeisA24356The 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! 🐙"I conjecture that $a(4)$ is the only zero. - Jon Perry, Mar 22 2004"
@[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