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 integral . Mahler (1968) 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.
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
-Problem.'' *Kongel. Norske Vidensk Selsk. Skr.* **16**, 1-4,
1972.

© 1996-9

1999-05-26