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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 part]]s

:<math>\left\lbrace x \left(\frac 3 2\right)^ n \right\rbrace </math>

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

:<math>\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). </math>

Mahler's conjecture would thus imply that Ω(3/2) exceeds 1/2.  Flatto, [[Jeffrey C. Lagarias|Lagarias]], and Pollington showed<ref>{{cite journal | first1=Leopold | last1=Flatto | first2=Jeffrey C. | last2=Lagarias | author2-link=Jeffrey C. Lagarias | first3=Andrew D. | last3=Pollington | title=On the range of fractional parts of&nbsp;''&zeta;''&nbsp;{&nbsp;(''p''/''q'')<sup>''n''</sup>&nbsp;} | journal=[[Acta Arithmetica]] | volume=LXX | number=2 | year=1995 | pages=125–147 | zbl=0821.11038 | issn=0065-1036 }}</ref> that

:<math>\Omega\left(\frac p q\right) > \frac 1 p </math>

for rational&nbsp;''p''/''q''.  

==References==
{{reflist}}
* {{cite book | last1=Everest | first1=Graham | last2=van der Poorten | first2=Alf | author2-link=Alf van der Poorten | last3=Shparlinski | first3=Igor | last4=Ward | first4=Thomas | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | location=[[Providence, RI]] | publisher=[[American Mathematical Society]] | year=2003 | isbn=0-8218-3387-1 | zbl=1033.11006 }}

[[Category:Analytic number theory]]
[[Category:Conjectures]]

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