Z-Number
A 
-number is a real number 
such that
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.
REFERENCES:
Flatto, L. "
-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.
