TOPICS
Search

Z-Number


A Z-number is a real number xi such that

 0<=frac[(3/2)^kxi]<1/2

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 integer n, and 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

Explore with Wolfram|Alpha

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.Mahler, K. "An Unsolved Problem on the Powers of 3/2." Austral. Math. Soc. 8, 313-321, 1968.Tijdman, R. "Note on Mahler's 3/2-Problem." Kongel. Norske Vidensk Selsk. Skr. 16, 1-4, 1972.

Referenced on Wolfram|Alpha

Z-Number

Cite this as:

Weisstein, Eric W. "Z-Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Z-Number.html

Subject classifications