Collatz Conjecture for Modulo an Integer

Abstract

A function $T_m$ from a set \{1,2,3,...$m$\} into itself defined by $T_m(x)=\frac{x}{2},$ for even $x$ and by $T_m(x)=\frac{3x+1}{2}$ (mod $m$), for odd $x$ is considered in this article. The asymptotic behaviour of this function is studied in this article for some cases.