Mahler's $\frac{3}{2}$ problem in ${\mathbb{Z}}^{+}$

This problem was asked to K. Mahler by one of his Japanese colleagues, a Z-number is a positive real number $x$ such that the fractional parts of $x(\frac{3}{2}{)}^n$ are less than $\frac{1}{2}$ for all integers $n$ such that $n\ge 0$. Kurt Mahler conjectured in 1968 that there are no Z-numbers. In this paper, we show that there are no Z-numbers in ${\mathbb{Z}}^{+}=\{1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}\}$.