Mahler's 3/2 problem

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

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

${\displaystyle x\left({\frac {3}{2}}\right)^{n}}$

are less than 1/2 for all positive integers n. Kurt Mahler conjectured in 1968 that there are no Z-numbers ^[1] .

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

${\displaystyle \Omega (\alpha )=\inf _{\theta >0}\left({\limsup _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace -\liminf _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace }\right).}$

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

${\displaystyle \Omega \left({\frac {p}{q}}\right)>{\frac {1}{p}}}$

for rational p/q > 1 in lowest terms.

References

1. Mahler, Kurt (1968). "An unsolved problem on the powers of 3/2". The Journal of the Australian Mathematical Society. VIII (2): 313–321. doi: 10.1017/S1446788700005371 . ISSN   1446-7887. Zbl   0155.09501.
2. 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. doi: 10.4064/aa-70-2-125-147 . ISSN   0065-1036. Zbl   0821.11038.

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