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In mathematics, Mahler's 3/2 problem concerns the existence of "Z-numbers".

A Z-number is a real number x such that the fractional parts

\( \left\lbrace x \left(\frac 3 2\right)^ n \right\rbrace \)

are less than 1/2 for all natural numbers n. Kurt Mahler conjectured in 1968 that there are no Z-numbers.

More generally, for a real number α, define Ω(α) as

\( \Omega(\alpha) = \inf_\theta\left({ \limsup_{n \rightarrow \infty} \left\lbrace{\theta\alpha^n}\right\rbrace - \liminf_{n \rightarrow \infty} \left\lbrace{\theta\alpha^n}\right\rbrace }\right). \)

Mahler's conjecture would thus imply that Ω(3/2) exceeds 1/2. Flatto, Lagarias, and Pollington showed[1] that

\( \Omega\left(\frac p q\right) > \frac 1 p \)

for rational p/q.
References

Flatto, Leopold; Lagarias, Jeffrey C.; Pollington, Andrew D. (1995). "On the range of fractional parts of ζ { (p/q)n }". Acta Arithmetica LXX (2): 125–147. ISSN 0065-1036. Zbl 0821.11038.

Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs 104. Providence, RI: American Mathematical Society. ISBN 0-8218-3387-1. Zbl 1033.11006.

Mathematics Encyclopedia

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