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import FormalConjecturesUtilGilbreath's conjecture
namespace Gilbreath
Gilbreath's nth difference, $d^n$ Let $d^0(n) = p_n$ and $d^k(n) = |d^{k-1}(n+1) - d^{k-1}(n)|
noncomputable def d : ℕ → (ℕ → ℕ)
| 0 => fun n ↦ n.nth Nat.Prime
| k + 1 => fun n ↦ Int.natAbs (d k (n + 1) - d k n)
open Gilbreath
Gilbreath's conjecture Gilbreath's conjecture states that every term in the sequence $d^k_0$ for $k > 0$ is equal to 1.
@[category research open, AMS 11]
theorem gilbreath_conjecture (k : ℕ+) : d k 0 = 1 := k:ℕ+⊢ d (↑k) 0 = 1
All goals completed! 🐙
end Gilbreath