FormalConjectures.OEIS.«78729»

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import FormalConjecturesUtil

Least $k > 0$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime

The sequence $a(n)$ is the least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime, if such $k$ exists; otherwise $a(n) = 0$.

References:

A078729

namespace OeisA78729open Classical in

Least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime, or 0 if no such $k$ exists.

noncomputable def a (n : ℕ) : ℕ :=
  if h : ∃ k, 0 < k ∧ (∏ i ∈ Finset.range n, (k + i + 1) + 1).Prime then
    Nat.find h
  else
    0

set_option backward.isDefEq.respectTransparency false in

Value of the sequence a at 1.


@[category test, AMS 11]
theorem a_1 : a 1 = 1 := by⊢ a 1 = 1
  classical
  -- `dsimp` normalises `Finset.range 1` inside the condition but not in the `Decidable`
  -- instance, which stops `split_ifs` from firing.
  dsimp [a]⊢ (if h : ∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1) then Nat.find h else 0) = 1
  split_ifs with hposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1)⊢ Nat.find h = 1negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1)⊢ False
  ·posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1)⊢ Nat.find h = 1 rw [Nat.find_eq_iffposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1)⊢ (0 < 1 ∧ Nat.Prime (∏ i ∈ {0}, (1 + i + 1) + 1)) ∧ ∀ n < 1, ¬(0 < n ∧ Nat.Prime (∏ i ∈ {0}, (n + i + 1) + 1))posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1)⊢ (0 < 1 ∧ Nat.Prime (∏ i ∈ {0}, (1 + i + 1) + 1)) ∧ ∀ n < 1, ¬(0 < n ∧ Nat.Prime (∏ i ∈ {0}, (n + i + 1) + 1))]posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1)⊢ (0 < 1 ∧ Nat.Prime (∏ i ∈ {0}, (1 + i + 1) + 1)) ∧ ∀ n < 1, ¬(0 < n ∧ Nat.Prime (∏ i ∈ {0}, (n + i + 1) + 1))
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1)⊢ False exact (h ⟨1, byh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ {0}, (k + i + 1) + 1)⊢ 0 < 1 ∧ Nat.Prime (∏ i ∈ {0}, (1 + i + 1) + 1) decide +nativeAll goals completed! 🐙⟩).elim

Value of the sequence a at 2.


@[category test, AMS 11]
theorem a_2 : a 2 = 1 := by⊢ a 2 = 1
  classical
  dsimp [a]⊢ (if h : ∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1) then Nat.find h else 0) = 1
  split_ifs with hposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1)⊢ Nat.find h = 1negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1)⊢ False
  ·posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1)⊢ Nat.find h = 1 rw [Nat.find_eq_iffposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1)⊢ (0 < 1 ∧ Nat.Prime (∏ i ∈ Finset.range 2, (1 + i + 1) + 1)) ∧
  ∀ n < 1, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 2, (n + i + 1) + 1))posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1)⊢ (0 < 1 ∧ Nat.Prime (∏ i ∈ Finset.range 2, (1 + i + 1) + 1)) ∧
  ∀ n < 1, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 2, (n + i + 1) + 1))]posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1)⊢ (0 < 1 ∧ Nat.Prime (∏ i ∈ Finset.range 2, (1 + i + 1) + 1)) ∧
  ∀ n < 1, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 2, (n + i + 1) + 1))
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1)⊢ False exact (h ⟨1, byh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 2, (k + i + 1) + 1)⊢ 0 < 1 ∧ Nat.Prime (∏ i ∈ Finset.range 2, (1 + i + 1) + 1) decide +nativeAll goals completed! 🐙⟩).elim

Value of the sequence a at 3.


@[category test, AMS 11]
theorem a_3 : a 3 = 2 := by⊢ a 3 = 2
  classical
  dsimp [a]⊢ (if h : ∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1) then Nat.find h else 0) = 2
  split_ifs with hposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1)⊢ Nat.find h = 2negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1)⊢ False
  ·posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1)⊢ Nat.find h = 2 rw [Nat.find_eq_iffposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 3, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 3, (n + i + 1) + 1))posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 3, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 3, (n + i + 1) + 1))]posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 3, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 3, (n + i + 1) + 1))
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1)⊢ False exact (h ⟨2, byh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 3, (k + i + 1) + 1)⊢ 0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 3, (2 + i + 1) + 1) decide +nativeAll goals completed! 🐙⟩).elim

Value of the sequence a at 5.


@[category test, AMS 11]
theorem a_5 : a 5 = 2 := by⊢ a 5 = 2
  classical
  dsimp [a]⊢ (if h : ∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1) then Nat.find h else 0) = 2
  split_ifs with hposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1)⊢ Nat.find h = 2negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1)⊢ False
  ·posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1)⊢ Nat.find h = 2 rw [Nat.find_eq_iffposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 5, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 5, (n + i + 1) + 1))posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 5, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 5, (n + i + 1) + 1))]posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 5, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 5, (n + i + 1) + 1))
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1)⊢ False exact (h ⟨2, byh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 5, (k + i + 1) + 1)⊢ 0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 5, (2 + i + 1) + 1) decide +nativeAll goals completed! 🐙⟩).elim

Value of the sequence a at 6.


@[category test, AMS 11]
theorem a_6 : a 6 = 2 := by⊢ a 6 = 2
  classical
  dsimp [a]⊢ (if h : ∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1) then Nat.find h else 0) = 2
  split_ifs with hposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1)⊢ Nat.find h = 2negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1)⊢ False
  ·posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1)⊢ Nat.find h = 2 rw [Nat.find_eq_iffposh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 6, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 6, (n + i + 1) + 1))posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 6, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 6, (n + i + 1) + 1))]posh:∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1)⊢ (0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 6, (2 + i + 1) + 1)) ∧
  ∀ n < 2, ¬(0 < n ∧ Nat.Prime (∏ i ∈ Finset.range 6, (n + i + 1) + 1))
    decide +nativeAll goals completed! 🐙
  ·negh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1)⊢ False exact (h ⟨2, byh:¬∃ k, 0 < k ∧ Nat.Prime (∏ i ∈ Finset.range 6, (k + i + 1) + 1)⊢ 0 < 2 ∧ Nat.Prime (∏ i ∈ Finset.range 6, (2 + i + 1) + 1) decide +nativeAll goals completed! 🐙⟩).elim

$(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2$, which is never prime. Hence $a(4) = 0$. Conjecture: $a(n) = 0$ if and only if $n = 4$.

@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 0 < n) : a n = 0 ↔ n = 4 := byn:ℕhn:0 < n⊢ a n = 0 ↔ n = 4
  sorryAll goals completed! 🐙

end OeisA78729