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import FormalConjecturesUtilLeast $k \ge 1$ such that $2 \cdot n^k - 1$ is prime
$a(n) = \min {k \ge 1 \mid \text{Prime}(2 \cdot n^k - 1)}$ for $n \ge 2$.
References:
namespace OeisA119591Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime, or $0$ if no such $k$ exists.
noncomputable def a (n : ℕ) : ℕ :=
sInf {k : ℕ | 0 < k ∧ (2 * n ^ k - 1).Prime}
Value of the sequence a at 0.
All goals completed! 🐙
Value of the sequence a at 1.
All goals completed! 🐙
Value of the sequence a at 2.
h_least:IsLeast {k | 0 < k ∧ Nat.Prime (2 * 2 ^ k - 1)} 1⊢ a 2 = 1
All goals completed! 🐙
Value of the sequence a at 3.
h_least:IsLeast {k | 0 < k ∧ Nat.Prime (2 * 3 ^ k - 1)} 1⊢ a 3 = 1
All goals completed! 🐙
Value of the sequence a at 4.
h_least:IsLeast {k | 0 < k ∧ Nat.Prime (2 * 4 ^ k - 1)} 1⊢ a 4 = 1
All goals completed! 🐙
Value of the sequence a at 5.
h_least:IsLeast {k | 0 < k ∧ Nat.Prime (2 * 5 ^ k - 1)} 4⊢ a 5 = 4
All goals completed! 🐙
Value of the sequence a at 6.
h_least:IsLeast {k | 0 < k ∧ Nat.Prime (2 * 6 ^ k - 1)} 1⊢ a 6 = 1
All goals completed! 🐙Is $a(n)$ defined for all $n \ge 2$? That is, does there exist $k > 0$ such that $2 \cdot n^k - 1$ is prime?
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 2 ≤ n) : ∃ k > 0, (2 * n ^ k - 1).Prime := n:ℕhn:2 ≤ n⊢ ∃ k > 0, Nat.Prime (2 * n ^ k - 1)
All goals completed! 🐙end OeisA119591