Convergence of generalized Collatz problem in k-adic field

Abstract

In this article, we define a new k-adic series transformation called Z-transformation and probe into its fixed point and periodicity. We extend the number field of the transform period problem to a wider k-adic field. Different constraints are imposed on k, then different periodic columns are formed after finite Z-transformations. We obtain that their periodic sequences are M1= {1,2} and M2={1,2}\cup {n_0}\cup {n'} respectively after derivation. As an application, it can provide a reference for C problems in more complex algebraic ystems.