Mahler's $\frac{3}{2}$ problem in $\mathbb{Z}^{+} $

Nikhil S Kumar

Mahler's $\frac{3}{2}$ problem in $\mathbb{Z}^{+} $

Number Theory 2025-06-19 v2

Authors: Nikhil S Kumar

Abstract

This problem was asked to K. Mahler by one of his Japanese colleagues, a Z-number is a positive real number $x$ such that the fractional parts of $x(\frac{3}{2})^n$ are less than $\frac{1}{2}$ for all integers $n$ such that $n \ge 0$ . Kurt Mahler conjectured in 1968 that there are no Z-numbers. In this paper, we show that there are no Z-numbers in $\mathbb{Z}^{+} = \{1,2,3,...\}$ .

Keywords

number theory

Cite

@article{arxiv.2411.03468,
  title  = {Mahler's $\frac{3}{2}$ problem in $\mathbb{Z}^{+} $},
  author = {Nikhil S Kumar},
  journal= {arXiv preprint arXiv:2411.03468},
  year   = {2025}
}

Comments

The paper has an alternate proof method, but there are trivial ways to prove the same

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