FormalConjectures.OEIS.«1223»

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module

public import FormalConjecturesUtil

Prime gaps

Differences between consecutive primes: $a(n) = p_{n+1} - p_n$.

References:

A001223

@[expose] public sectionnamespace OeisA1223

Prime gaps: differences between consecutive primes.

noncomputable def a (n : ℕ) : ℕ :=
  if n = 0 then 0
  else Nat.nth Nat.Prime n - Nat.nth Nat.Prime (n - 1)

Helper definition for extracting a finite subsequence (pattern) as a list.

noncomputable def gapSubsequence (startIndex length : ℕ) : List ℕ :=
  (List.range length).map (fun i => a (startIndex + i))

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 = 1 := by⊢ a 1 = 1
  change Nat.nth Nat.Prime 1 - Nat.nth Nat.Prime 0 = 1⊢ Nat.nth Nat.Prime 1 - Nat.nth Nat.Prime 0 = 1
  rw [Nat.nth_prime_one_eq_three,⊢ 3 - Nat.nth Nat.Prime 0 = 1All goals completed! 🐙 Nat.nth_prime_zero_eq_two⊢ 3 - 2 = 1All goals completed! 🐙]All goals completed! 🐙

Value of the sequence a at 2.


@[category test, AMS 11]
theorem a_2 : a 2 = 2 := by⊢ a 2 = 2
  change Nat.nth Nat.Prime 2 - Nat.nth Nat.Prime 1 = 2⊢ Nat.nth Nat.Prime 2 - Nat.nth Nat.Prime 1 = 2
  rw [Nat.nth_prime_two_eq_five,⊢ 5 - Nat.nth Nat.Prime 1 = 2All goals completed! 🐙 Nat.nth_prime_one_eq_three⊢ 5 - 3 = 2All goals completed! 🐙]All goals completed! 🐙

Value of the sequence a at 3.


@[category test, AMS 11]
theorem a_3 : a 3 = 2 := by⊢ a 3 = 2
  change Nat.nth Nat.Prime 3 - Nat.nth Nat.Prime 2 = 2⊢ Nat.nth Nat.Prime 3 - Nat.nth Nat.Prime 2 = 2
  rw [Nat.nth_prime_three_eq_seven,⊢ 7 - Nat.nth Nat.Prime 2 = 2All goals completed! 🐙 Nat.nth_prime_two_eq_five⊢ 7 - 5 = 2All goals completed! 🐙]All goals completed! 🐙

Any subsequence a(n .. n+m) with n > 2 (as to exclude the untypical primes 2 and 3) should occur infinitely many times at other starting points k.

This is false. The five-term block starting at $n = 3$ is $(2,4,2,4,2)$, and a congruence modulo $5$ shows that it occurs only at $n = 3$.

@[category research solved, AMS 11, formal_proof using lean4 at
  "https://github.com/epoch-research/LeanOpenProblems-results/blob/f02efd9a8c5fc6a735d2a90c33e24f7278ce0ffc/runs/oeis-full-50usd-oai-jajpvieznaevpoyg/prime_gap_subsequences_occur_infinitely_often/Submission/Spec.lean#L134"]
theorem conjecture :
    ¬ ∀ (n m : ℕ), n ≥ 3 →
      Set.Infinite {k : ℕ | gapSubsequence k (m + 1) = gapSubsequence n (m + 1)} := by⊢ ¬∀ (n m : ℕ), n ≥ 3 → {k | gapSubsequence k (m + 1) = gapSubsequence n (m + 1)}.Infinite
  sorryAll goals completed! 🐙

end OeisA1223