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import FormalConjecturesUtilRiesel Problem
Riesel problem: let $k=2n-1$; then $a(n)$ is the smallest $m \ge 1$ such that $k \cdot 2^m-1$ is prime, or $-1$ if no such prime exists.
References:
namespace OeisA108129variable {m n : ℕ}open Natopen Classical in
The primary defining sequence a.
Riesel problem: let $k=2n-1$; then $a(n)$ is the smallest $m \ge 1$ such that
$k \cdot 2^m-1$ is prime, or $-1$ if no such prime exists.
noncomputable def a (n : ℕ) : ℤ :=
if n = 0 then 0
-- Use classical choice to find the minimum, or return -1 if no such prime exists.
else if h : ∃ m, m ≠ 0 ∧ ((2 * n - 1) * 2 ^ m - 1).Prime then
Nat.find h
else
-1All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 2 := a_of_isLeast <| ⊢ IsLeast {m | m ≠ 0 ∧ Nat.Prime ((2 * 1 - 1) * 2 ^ m - 1)} 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 1 := a_of_isLeast <| ⊢ IsLeast {m | m ≠ 0 ∧ Nat.Prime ((2 * 2 - 1) * 2 ^ m - 1)} 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 2 := a_of_isLeast <| ⊢ IsLeast {m | m ≠ 0 ∧ Nat.Prime ((2 * 3 - 1) * 2 ^ m - 1)} 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 1 := a_of_isLeast <| ⊢ IsLeast {m | m ≠ 0 ∧ Nat.Prime ((2 * 4 - 1) * 2 ^ m - 1)} 1 All goals completed! 🐙It is conjectured that the integer $k = 509203$ is the smallest Riesel number, that is, the first $n$ such that $a(n) = -1$ is $254602$.
@[category research open, AMS 11]
theorem conjecture :
a 254602 = -1 ∧ (∀ n : ℕ, 1 ≤ n ∧ n < 254602 → a n ≠ -1) := ⊢ a 254602 = -1 ∧ ∀ (n : ℕ), 1 ≤ n ∧ n < 254602 → OeisA108129.a n ≠ -1
All goals completed! 🐙end OeisA108129