Rowland-style prime-generating recurrence

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import FormalConjecturesUtil

/-!
# Rowland-style prime-generating recurrence

The sequence is defined by $a(1) = 2$ and for $n \ge 2$:
$a(n) = a(n-1) + \gcd(n, a(n-1))$ if $n$ is even, and
$a(n) = a(n-1) + \gcd(n-2, a(n-1))$ if $n$ is odd.

*References:*
- [A166944](https://oeis.org/A166944)
- E. S. Rowland, "A natural prime-generating recurrence", *J. Integer Sequences* **11** (2008),
  Article 08.2.8.
- V. Shevelev, "An infinite set of generators of primes based on the Rowland idea and
  conjectures concerning twin primes", arXiv preprint
  [arXiv:0910.4676](https://arxiv.org/abs/0910.4676) [math.NT], 2009.-/

namespace OeisA166944

/-- Defining recurrence for $a(n)$. -/
def a : ℕ → ℕ
  | 0 => 0
  | 1 => 2
  | n + 2 =>
    let prev := a (n + 1)
    let idx := n + 2
    if idx % 2 = 0 then prev + Nat.gcd idx prev
    else prev + Nat.gcd (idx - 2) prev

/-- Value of the sequence `a` at 1. -/
@[category test, AMS 11]
theorem a_1 : a 1 = 2 := by decide

/-- Value of the sequence `a` at 2. -/
@[category test, AMS 11]
theorem a_2 : a 2 = 4 := by decide

/-- Value of the sequence `a` at 3. -/
@[category test, AMS 11]
theorem a_3 : a 3 = 5 := by decide

/-- Value of the sequence `a` at 4. -/
@[category test, AMS 11]
theorem a_4 : a 4 = 6 := by decide

/-- Value of the sequence `a` at 5. -/
@[category test, AMS 11]
theorem a_5 : a 5 = 9 := by decide

/-- The difference sequence $d(n) = a(n) - a(n-1)$ for $n \ge 2$. -/
def d (n : ℕ) : ℕ := a n - a (n - 1)

/-- A prime $p$ is the greater of a twin prime pair if $p$ and $p - 2$ are both prime. -/
def IsGreaterTwinPrime (p : ℕ) : Prop := p.Prime ∧ (p - 2).Prime

/--
A value $R$ is a difference record if there exists $n \ge 2$ such that $d(n) = R$ and $R$
is strictly larger than all previous differences $d(k)$ for $2 \le k < n$.-/
def IsDifferenceRecord (R : ℕ) : Prop :=
  ∃ n : ℕ, 2 ≤ n ∧ d n = R ∧ ∀ k : ℕ, 2 ≤ k → k < n → d k < R

/--
Conjecture: Every record of differences $a(n)-a(n-1)$ more than 5 is the greater of twin primes
(A006512).-/
@[category research open, AMS 11]
theorem conjecture (R : ℕ) (hR : 5 < R) (hrec : IsDifferenceRecord R) :
    IsGreaterTwinPrime R := by
  sorry

end OeisA166944

• Formalization source ↗

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