A Topological Path to the Collatz Conjecture: A historical point of view

Abstract

In this paper a recent topological approach to the Collatz conjecture is shown. Also known as “the 3 x + 1 problem”, the Collatz conjecture establishes that for every natural number n , there exists a natural number r such that the r -fold composite κ r ( n ) = 1 , where κ : N → N is the function defined as x/ 2 if x is even and as 3 x + 1 if x is odd. This map κ induces the so called primal topology τ κ on N . We prove that the space ( N , τ κ ) is path-connected and compact if and only if the Collatz conjecture is true.