Mahler's 3/2 problem
Appearance
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 of
are less than 1/2 for all positive integers n. Kurt Mahler conjectured in 1968 that there are no Z-numbers.
More generally, for a real number α, define Ω(α) as
Mahler's conjecture would thus imply that Ω(3/2) exceeds 1/2. Flatto, Lagarias, and Pollington showed[1] that
for rational p/q > 1 in lowest terms.
References
[edit]- ^ 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.