TOPICS
A 
-number is a real
number 
such that
![0<=frac[(3/2)^kxi]<1/2](/images/equations/Z-Number/NumberedEquation1.svg) |
for all 
, 2, ..., where frac
is the fractional part of 
. Mahler (1968) showed that there is at most one 
-number in each interval 
for integer 
, and therefore concluded that it is unlikely that any 
-numbers exist. The 
-numbers arise in the analysis of the Collatz
problem.
-Numbers and 
-Transformations." Symbolic Dynamics and its Applications,
Contemporary Math. 135, 181-201, 1992.Guy, R. K. "Mahler's
-Numbers." §E18 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 220,
1994.Lagarias, J. C. "The 
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 
-Problem."
Kongel. Norske Vidensk Selsk. Skr. 16, 1-4, 1972.Weisstein, Eric W. "Z-Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Z-Number.html