A TOPOLOGICAL APPROACH TO THE ULAM-KAKUTANI COLLATZ CONJECTURE

Abstract

The Ulam-Kakutani Collatz Conjecture The Ulam-Kakutani Collatz conjecture explores the behavior of sequences generated by the Collatz function, defined as follows: For any positive integer n, f(n) = 8><>: n2 ; if n is even; 3n + 1; if n is odd: A topological approach to this conjecture involves analyzing the set of integers as a discrete space and examining the dynamics of the function through its iterative mappings. The key aspects of this approach include: Mapping and fixed points, Topological dynamics, Compactness, Connectedness, and Orbit structure. This approach aims to provide insights into the underlying structure of the conjecture and its potential resolution through topological methods. In addition to this, the researcher proposes to prove that if (N; τf) is not a w-Ro space, it is equivalent to saying that the conjecture is true. Here, τf is the topology on N given by the open sets as those subsets θ of N such that f−1(θ) ⊂ θ, where f is the Collatz function.