Collatz Conjecture: Generalizing the Odd Part

Abstract

Our aim is to investigate the Collatz conjecture. Because the chaotic mixing from iterating the piecewise Collatz function takes place in the odd case, we restrict attention to the odd integers in the orbits to identify some regularities. The parity sequence is reinterpreted and then used to show that if a counterexample exists then there are infinitely many counterexamples with any given initial behavior. When replacing the subfunction 3x+1 in the odd case with other affine functions, our results generalize. We show that the prime factorizations of the coefficients can be used to put a lower bound on the number of weak components in the digraph generated. Furthermore, we identify pairs of functions in this class such that the graph generated by one is isomorphic to a subgraph of the graph generated by the other. In the end, the Collatz conjecture is generalized and several new conjectures are raised.