## Reprint: An unsolved problem on the powers of $3/2$ (1968)

##### Doc. Math. Extra Vol. Mahler Selecta, 639-648 (2019)
DOI: 10.25537/dm.2019.SB-639-648

### Summary

One says that $\alpha>0$ is a $Z$-number if $0\le \{\alpha (3/2)^n\}<1/2$, where $\{x\}$ denotes the fractional part of $x$. In this paper, while not showing existence, Mahler proves that the set of $Z$-numbers is at most countable. More specifically, Mahler shows that, up to $x$, there are at most $x^{0.7} Z$-numbers. \par Reprint of the author's paper [J. Aust. Math. Soc. 8, 313--321 (1968; Zbl 0155.09501)].

11-03, 11J54