Paths in primal spaces and the Collatz conjecture

Abstract

The Collatz conjecture establishes that for every natural number n ∈ ℕ, there exists an r ∈ ℕ such that κ^r (n) = 1, where κ : ℕ → ℕ is the function defined as n/2 if n es even and as 3n + 1 if n is odd. The map κ induces a topology τκ on ℕ. We prove that the Collatz conjecture and connectedness of the space (ℕ, τ_κ ) imply that the space is simply connected. Furthermore, we prove that the Collatz conjecture is equivalent to the fact that the space (ℕ, τ_κ ) is path-connected.