On the solution of the Collatz problem and on rational solutions of the   equations of elliptic curves
  Part one:   In this paper, we propose a key idea and innovation on the Collatz problem. Let an odd Collatz's Sequence be \[ S_{c}(m,t)=\{n_{0}(m,t),n_{1}(m,t),n_{2}(m,t),\ldots,n_{m}(m,t),n_{m+1}(m,t)\} \] produced by a function $n_{i+1}(m,t)=(3n_{i}(m,t)+1)/2$ and ended by an even number $n_{m+1}(m,t)$ where \[ n_{i}(m,t)=2^{m+1-i}\times3^{i}t-1\texttt{ for }i=0,1,2,\ldots,m+1 \] and the positive integer $t$ is odd and not divisible by $3$.   Then in two ways we prove that each Collatz's Sequence always returns to 1, i.e., the Collatz problem is solved.   Part two:   Given an equation $y^{2}=x^{3}+ax+b$ where $a\in Q\texttt{ and }b\in Q$, we prove that its pairs of rational solutions exist and can be determined only in some cases, each pair of which are $x$-axially symmetric, and the number of its pairs of rational solutions except for $x=0$ or $y=0$ or except for $x=\pm x_{0}$ or $y=0$ where $x_{0}\in Q$ is also proved equal to one or equal to two or equal to the number of plus cubic rational numbers only in some cases, and equal to zero in other cases. 
Shan-Guang Tan
math.GM