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 k^r(n) = 1, where k : ℕ → ℕ  is the function  defined as n / 2 if n es even and as 3n + 1 if n is odd. The map k induces a topology ?_k on ℕ. We prove that the Collatz conjecture and connectedness of the space (ℕ,?_k)imply that the space is simply connected. Furthermore, we prove that the Collatz conjecture is equivalent to the fact that the space (ℕ,?_k is path-connected.