1. FormalConjectures
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  3. «117027»
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Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes

$a(n) = p_{4n-3}p_{4n} - p_{4n-2}p_{4n-1}$ where $p_k$ is the k-th prime number (1-indexed).

References:

namespace OeisA117027open Nat Int Filter

a n is the determinant of a 2x2 matrix of non-overlapping blocks of 4 consecutive primes.

noncomputable def a (n : ℕ) : ℤ := if 0 < n then let k := 4 * n let pPrime (i : ℕ) : ℤ := (Nat.nth Nat.Prime i : ℤ) let p₁ := pPrime (k - 4) -- p_{4n-4} in 0-indexed Mathlib let p₂ := pPrime (k - 1) -- p_{4n-1} in 0-indexed Mathlib let p₃ := pPrime (k - 3) -- p_{4n-3} in 0-indexed Mathlib let p₄ := pPrime (k - 2) -- p_{4n-2} in 0-indexed Mathlib p₁ * p₂ - p₃ * p₄ else 0h1:Nat.Prime 13⊢ nth Nat.Prime 5 = 13 All goals completed! 🐙h1:Nat.Prime 17⊢ nth Nat.Prime 6 = 17 All goals completed! 🐙h1:Nat.Prime 19⊢ nth Nat.Prime 7 = 19 All goals completed! 🐙h1:Nat.Prime 23⊢ nth Nat.Prime 8 = 23 All goals completed! 🐙h1:Nat.Prime 29⊢ nth Nat.Prime 9 = 29 All goals completed! 🐙h1:Nat.Prime 31⊢ nth Nat.Prime 10 = 31 All goals completed! 🐙h1:Nat.Prime 37⊢ nth Nat.Prime 11 = 37 All goals completed! 🐙@[category test, AMS 11] theorem a_0 : a 0 = 0 := ⊢ a 0 = 0 All goals completed! 🐙⊢ ↑2 * ↑7 - ↑3 * ↑5 = -1 All goals completed! 🐙⊢ ↑11 * ↑19 - ↑13 * ↑17 = -12 All goals completed! 🐙⊢ ↑23 * ↑37 - ↑29 * ↑31 = -48 All goals completed! 🐙

The count of positive terms among $a(1)$, ..., $a(N)$.

noncomputable def positiveCount (N : ℕ) : ℕ := (List.range N).countP (fun n => 0 < a (n + 1))

The count of negative terms among $a(1)$, ..., $a(N)$.

noncomputable def negativeCount (N : ℕ) : ℕ := (List.range N).countP (fun n => a (n + 1) < 0)

The sequence of ratios $P(N)/Neg(N)$ as a sequence of real numbers.

noncomputable def ratioSeq (N : ℕ) : ℝ := if negativeCount N = 0 then 0 else (positiveCount N : ℝ) / (negativeCount N : ℝ)

This suggests the ratio is approaching a limit close to 0.87.

Formalized as: The sequence of ratios $P(N)/Neg(N)$ converges to a limit L, and L is in the interval (0.8, 0.9).

@[category research open, AMS 11] theorem conjecture : ∃ L : ℝ, Tendsto ratioSeq atTop (nhds L) ∧ 0.8 < L ∧ L < 0.9 := ⊢ ∃ L, Tendsto ratioSeq atTop (nhds L) ∧ 0.8 < L ∧ L < 0.9 All goals completed! 🐙end OeisA117027