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.