On the solution of the Collatz problem

Abstract: In this paper, we first prove that given a non-negative 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 in two ways we prove that each Collatz's Sequence always returns to 1, i.e., the Collatz problem is solved.