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 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.