Mahler's 3/2 Problem

Definitions and mathematical statement

For a real number $y$, let $\lfloor y\rfloor$ be its greatest integer not exceeding $y$, and let $\{y\}=y-\lfloor y\rfloor\in[0,1)$ be its fractional part. The literal question in the supplied statement is \[\exists x\in{\mathbb{R}}\ \ \forall n\in{\mathbb{Z}}_{\geq1},\qquad \left\{x\left(\frac32\right)^n\right\}<\frac12.\] The strict inequality and the starting index $n=1$ are retained. No condition $x>0$ is present in this literal statement.

The supplied literature triage checked 2026-08-17 records the affirmative witness $x=0$, with all displayed fractional parts equal to zero, and therefore labels this exact record solved / SOLVED-IN-LITERATURE. That source-provided observation must not be reported as a resolution of Mahler's standard positive-real problem. The corrected existence question identified by the review adds $x>0$ to the same fractional-part condition; the background attributes to Kurt Mahler in the 1960s the conjectural nonexistence of such positive real numbers, called $Z$-numbers in the supplied references.

For the positive-real problem the review reports Mahler's result that at most one $Z$-number lies in each unit interval and later scarcity and distribution restrictions; it reports the standard existence question as still open. These are supplied dated assessments. The registry's separate status field is open, reflecting a conflict between the broad historical label and the unrestricted literal wording; the two targets are kept distinct.

Short English statement

The stored question allows every real $x$ and is reported answered affirmatively by $x=0$. Mahler's intended problem requires $x>0$; the supplied review says that positive-real existence question remains open. The omitted positivity condition is therefore essential.

Sources

• Local UnsolvedMath registry, permanent ID 1244, source code catalogue entry, "Mahler's 3/2 Problem"; supplied statement, historical background, and corrective literature triage.
• K. Mahler, "An unsolved problem on the powers of 3/2," Journal of the Australian Mathematical Society 8 (1968), 313--321: https://doi.org/10.1017/S144678870000613X. Supplied primary source for the positive-real problem.
• A. Dubickas, "On the fractional parts of rational powers," Acta Arithmetica 141 (2010), 103--110: https://doi.org/10.4064/aa141-2-1. The supplied review says it defines a $Z$-number with positive parameter and treats nonexistence as Mahler's conjecture.
• S. Akiyama et al., "Rational self-affine tiles," American Mathematical Monthly 129 (2022), 537--555: https://doi.org/10.1080/00029890.2022.2061281. Supplied specialist secondary exposition reporting the positive-real question open.
• Supplied literature triage checked 2026-08-17, SOLVED-IN-LITERATURE for the unrestricted exact wording only; it explicitly prohibits treating this as a solution to the positive-real problem and cautions that dated triage does not exclude later or unindexed work.