FormalConjectures.OEIS.«1359»

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import FormalConjecturesUtil

Lesser of twin primes

Primes $p$ such that $p+2$ is also prime.

References:

A001359

namespace OeisA1359

The $n$-th lesser twin prime, with $a(0) = 0$.

noncomputable def a (n : ℕ) : ℕ :=
  if n > 0 then
    Nat.nth (fun p => p.Prime ∧ (p + 2).Prime) (n - 1)
  else
    0

Value of the sequence a at 0.

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

Value of the sequence a at 1.

@[category test, AMS 11]
theorem a_1 : a 1 = 3 := by⊢ a 1 = 3
  norm_num[a]⊢ Nat.nth (fun p ↦ Nat.Prime p ∧ Nat.Prime (p + 2)) 0 = 3
  exact(((congr_arg _) (by⊢ 0 = Nat.count (fun p ↦ Nat.Prime p ∧ Nat.Prime (p + 2)) 3 constructorAll goals completed! 🐙) )).trans.comp (3).nth_count (by⊢ Nat.Prime 3 ∧ Nat.Prime (3 + 2) decideAll goals completed! 🐙)

Value of the sequence a at 2.

@[category test, AMS 11]
theorem a_2 : a 2 = 5 := by⊢ a 2 = 5
  delta a⊢ (if 2 > 0 then Nat.nth (fun p ↦ Nat.Prime p ∧ Nat.Prime (p + 2)) (2 - 1) else 0) = 5
  apply((congr_arg _) (by⊢ 2 - 1 = Nat.count (fun p ↦ Nat.Prime p ∧ Nat.Prime (p + 2)) 5 constructorAll goals completed! 🐙) ).trans (Nat.nth_count (by⊢ Nat.Prime 5 ∧ Nat.Prime (5 + 2) decideAll goals completed! 🐙 ) )

Value of the sequence a at 3.

@[category test, AMS 11]
theorem a_3 : a 3 = 11 := by⊢ a 3 = 11
  (inhabit ℝinhabited_h:Inhabited ℝ⊢ a 3 = 11)
  norm_num[a]inhabited_h:Inhabited ℝ⊢ Nat.nth (fun p ↦ Nat.Prime p ∧ Nat.Prime (p + 2)) 2 = 11
  exact (congr_arg _ (byinhabited_h:Inhabited ℝ⊢ 2 = Nat.count (fun p ↦ Nat.Prime p ∧ Nat.Prime (p + 2)) 11 decideAll goals completed! 🐙)).trans (Nat.nth_count (byinhabited_h:Inhabited ℝ⊢ Nat.Prime 11 ∧ Nat.Prime (11 + 2) decideAll goals completed! 🐙))

Value of the sequence a at 4.

@[category test, AMS 11]
theorem a_4 : a 4 = 17 := by⊢ a 4 = 17
  simp_all[a]⊢ Nat.nth (fun p ↦ Nat.Prime p ∧ Nat.Prime (p + 2)) 3 = 17
  exact (congr_arg _ (by⊢ 3 = Nat.count (fun p ↦ Nat.Prime p ∧ Nat.Prime (p + 2)) 17 constructorAll goals completed! 🐙) ).trans (Nat.nth_count (by⊢ Nat.Prime 17 ∧ Nat.Prime (17 + 2) decideAll goals completed! 🐙))

Primes $p_k$ such that $p_k! \equiv 1 \pmod{p_{k+1}}$ with the exception of $p_{991} = 7841$ and other unknown primes $p_k$ for which $(p_k+1)(p_k+2)\cdots(p_{k+1}-2) \equiv 1 \pmod{p_{k+1}}$ where $p_{k+1} - p_k > 2$.

@[category research open, AMS 11]
theorem conjecture (k : ℕ) (hk : k > 1) :
    let Pk := Nat.nth Nat.Prime (k - 1)
    let Pk_succ := Nat.nth Nat.Prime k
    let Congruence := Pk.factorial ≡ 1 [MOD Pk_succ]
    let IsLesserTwinPrime := (Pk + 2).Prime
    let Wk_prod : ℕ := ∏ i ∈ Finset.Icc (Pk + 1) (Pk_succ - 2), i
    Congruence ↔ (IsLesserTwinPrime ∨ (k = 991) ∨ (Pk_succ - Pk > 2 ∧ Wk_prod ≡ 1 [MOD Pk_succ])) :=
        byk:ℕhk:k > 1⊢ let Pk := Nat.nth Nat.Prime (k - 1);
let Pk_succ := Nat.nth Nat.Prime k;
let Congruence := Pk.factorial ≡ 1 [MOD Pk_succ];
let IsLesserTwinPrime := Nat.Prime (Pk + 2);
let Wk_prod := ∏ i ∈ Finset.Icc (Pk + 1) (Pk_succ - 2), i;
Congruence ↔ IsLesserTwinPrime ∨ k = 991 ∨ Pk_succ - Pk > 2 ∧ Wk_prod ≡ 1 [MOD Pk_succ]
  sorryAll goals completed! 🐙

end OeisA1359