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

TL;DR

This paper proves that there are no positive integers that satisfy Mahler's Z-number condition, confirming Mahler's conjecture for the set of positive integers.

Contribution

The paper provides a proof that Mahler's Z-number conjecture holds for all positive integers, resolving a long-standing open problem.

Findings

No positive integers are Z-numbers.

Mahler's conjecture is confirmed for $ extbf{Z}^+$.

The result advances understanding of fractional parts in exponential sequences.

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 $\mathbb{Z}^{+} = \{1,2,3,...\}$.