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import FormalConjecturesUtil
namespace OeisA1146
The primary defining sequence a.
$a(n) = 2^{2^n}$.
def a (n : ℕ) : ℕ := 2 ^ (2 ^ n)@[category test, AMS 11]
theorem a_0 : a 0 = 2 := ⊢ a 0 = 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 4 := ⊢ a 1 = 4 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 16 := ⊢ a 2 = 16 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 256 := ⊢ a 3 = 256 All goals completed! 🐙@[category API, AMS 11]
lemma n_add_two_le_two_pow (n : ℕ) (hn : 2 ≤ n) : n + 2 ≤ 2 ^ n := n:ℕhn:2 ≤ n⊢ n + 2 ≤ 2 ^ n
n:ℕ⊢ 2 + 2 ≤ 2 ^ 2n:ℕk:ℕhk:2 ≤ kih:k + 2 ≤ 2 ^ k⊢ k + 1 + 2 ≤ 2 ^ (k + 1)
n:ℕ⊢ 2 + 2 ≤ 2 ^ 2 All goals completed! 🐙
n:ℕk:ℕhk:2 ≤ kih:k + 2 ≤ 2 ^ k⊢ k + 1 + 2 ≤ 2 ^ (k + 1) calc
k + 1 + 2 = k + 2 + 1 := n:ℕk:ℕhk:2 ≤ kih:k + 2 ≤ 2 ^ k⊢ k + 1 + 2 = k + 2 + 1 All goals completed! 🐙
_ ≤ 2 ^ k + 1 := n:ℕk:ℕhk:2 ≤ kih:k + 2 ≤ 2 ^ k⊢ k + 2 + 1 ≤ 2 ^ k + 1 All goals completed! 🐙
_ ≤ 2 ^ k + 2 ^ k := n:ℕk:ℕhk:2 ≤ kih:k + 2 ≤ 2 ^ k⊢ 2 ^ k + 1 ≤ 2 ^ k + 2 ^ k
n:ℕk:ℕhk:2 ≤ kih:k + 2 ≤ 2 ^ k⊢ 1 ≤ 2 ^ k
All goals completed! 🐙
_ = 2 ^ (k + 1) := n:ℕk:ℕhk:2 ≤ kih:k + 2 ≤ 2 ^ k⊢ 2 ^ k + 2 ^ k = 2 ^ (k + 1) All goals completed! 🐙The forward direction: if $k = a(n)$ for $n \ge 2$, then $k^4 - 1$ divides $2^k - 1$.
n:ℕhn:2 ≤ nm:ℕ := 2 ^ (2 ^ n - (n + 2))h_exp:2 ^ 2 ^ n = 2 ^ n * 4 * mH2:2 ^ a n = (a n ^ 4) ^ m⊢ a n ^ 4 - 1 ∣ (a n ^ 4) ^ m - 1
All goals completed! 🐙I conjecture that { $a(n)$ ; $n>1$ } are the numbers such that $n^4-1$ divides $2^n-1$, intersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015 This formalizes the reverse direction.
@[category research open, AMS 11]
theorem conjecture :
∀ k : ℕ, ((k^4 - 1) : ℕ) ∣ (2^k - 1 : ℕ) → k > 1 → ∃ n : ℕ, 2 ≤ n ∧ k = a n := ⊢ ∀ (k : ℕ), k ^ 4 - 1 ∣ 2 ^ k - 1 → k > 1 → ∃ n, 2 ≤ n ∧ k = OeisA1146.a n
All goals completed! 🐙end OeisA1146