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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 κ<sup>r</sup> (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<inf>κ</inf>) imply that the space is simply connected. Furthermore, we prove that the Collatz conjecture is equivalent to the fact that the space (ℕ, τ<inf>κ</inf>) is path-connected.