Z-Number

A $Z$-number is a Real Number $\xi$ such that

$$0\leq \mathop{\rm frac}\left[{\left({3\over 2}\right)^k\xi}\right]< {\textstyle{1\over 2}}$$

for all $k=1$, 2, ..., where frac$(x)$ is the fractional part of $x$. Mahler (1968) showed that there is at most one $Z$-number in each interval $[n, n+1)$ for integral $n$. Mahler (1968) therefore concluded that it is unlikely that any $Z$-numbers exist. The $Z$-numbers arise in the analysis of the Collatz Problem.

See also Collatz Problem

References

Flatto, L. ``$Z$-Numbers and $\beta$-Transformations.'' Symbolic Dynamics and its Applications, Contemporary Math. 135, 181-201, 1992.

Guy, R. K. ``Mahler's $Z$-Numbers.'' §E18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 220, 1994.

Lagarias, J. C. ``The $3x+1$ Problem and its Generalizations.'' Amer. Math. Monthly 92, 3-23, 1985. http://www.cecm.sfu.ca/organics/papers/lagarias/.

Mahler, K. ``An Unsolved Problem on the Powers of 3/2.'' Austral. Math. Soc. 8, 313-321, 1968.

Tijdman, R. ``Note on Mahler's ${\textstyle{3\over 2}}$-Problem.'' Kongel. Norske Vidensk Selsk. Skr. 16, 1-4, 1972.