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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 κr (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, τκ) imply that the space is simply connected. Furthermore, we prove that the Collatz conjecture is equivalent to the fact that the space (ℕ, τκ) is path-connected.