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import FormalConjecturesUtilDeterminant 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:
namespace OeisA71524open MatrixDeterminant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^2 + (j+1)^2$ is prime, and $0$ otherwise.
def a (n : ℕ) : ℤ :=
let M : Matrix (Fin n) (Fin n) ℤ := fun i j =>
let i_idx : ℕ := i.val + 1
let j_idx : ℕ := j.val + 1
if (i_idx ^ 2 + j_idx ^ 2).Prime then 1 else 0
M.det
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1
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 3.
@[category test, AMS 11]
theorem a_3 : a 3 = -1 := ⊢ a 3 = -1
All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 1 := ⊢ a 4 = 1
All goals completed! 🐙
Value of the sequence a at 5.
@[category test, AMS 11]
theorem a_5 : a 5 = 1 := ⊢ a 5 = 1
All goals completed! 🐙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.
def generalDet (m n : ℕ) : ℤ :=
let M : Matrix (Fin n) (Fin n) ℤ := fun i j =>
let i_idx : ℕ := i.val + 1
let j_idx : ℕ := j.val + 1
if (i_idx ^ (2 ^ m) + j_idx ^ (2 ^ m)).Prime then 1 else 0
M.detConjecture: $a(n) = 0$ for no $n > 28$.
Zhi-Wei Sun, Aug 26 2013
@[category research open, AMS 11 15]
theorem conjecture1 (n : ℕ) (hn : 28 < n) : a n ≠ 0 := n:ℕhn:28 < n⊢ a n ≠ 0
All goals completed! 🐙
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
@[category research open, AMS 11 15]
theorem conjecture2 (m : ℕ) : ∃ N : ℕ, ∀ n : ℕ, N < n → generalDet m n ≠ 0 := m:ℕ⊢ ∃ N, ∀ (n : ℕ), N < n → generalDet m n ≠ 0
All goals completed! 🐙end OeisA71524