On the solution of the Collatz problem

Abstract

In this paper, we first prove that given a nonnegative integer $m$ and an odd number $t$ not divisible by $3$, there exists a unique Collatz's Sequence \[ S_{c}(m,t)=\{n_{0}(m,t),n_{1}(m,t),n_{2}(m,t),\ldots,n_{m}(m,t),n_{m+1}(m,t)\} \] produced by a function $n_{i+1}(m,t)=(3n_{i}(m,t)+1)/2$ for $i=0,1,2,\ldots,m$ and ended by an even number $n_{m+1}(m,t)$ where $n_{i}(m,t)=2^{m+1-i}\times3^{i}t-1$ for $i=0,1,2,\ldots,m+1$, by which all odd numbers can be expressed. Then we discuss the Collatz problem in two ways and prove that each Collatz's Sequence always returns to 1, i.e., the Collatz problem is solved.