Excluded fractional-part windows for powers of 3/2
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Can all fractional parts of ξ(3/2)ⁿ stay in a fixed interval for every n≥0? Mahler’s Z-number problem uses positive ξ and [0,1/2); the recorded results exclude other specific single windows.
Why it matters and what counts as progress
Why it matters: A larger excluded window strengthens quantitative orbit-spreading information without necessarily resolving the original Z-number interval.
What counts as progress: A sound finite-block certificate excluding a new specified window, with endpoints and the range of ξ explicit, or a confined orbit meeting those exact conditions.
How the problem developed
- 1968 · Problem [1]
Kurt Mahler Posed the Z-number problem: whether some positive ξ has every fractional part of ξ(3/2)ⁿ in [0,1/2).
- 2008 · Partial result [2]
Artūras Dubickas Proved explicit confinement exclusions for powers of 3/2, including a union of two windows; this did not settle Mahler’s full interval question.
- 8 Aug 2026 · Formalized result [3]
Ralf Stephan; Claude Fable 5 supplied the mathematical discovery and Claude Opus 4.8 the Lean formalization Certified that no positive ξ keeps all fractional parts in the specific half-open interval [1/6,13/24); also formalized additional window and union exclusions.
AI contributions 2
Confinement schemas and the reach of block certificates for powers of rational numbers modulo one
For every nonzero real ξ, the fractional parts of ξ(3/2)ⁿ cannot all lie in the closed interval [2/7,5/7]. Equivalently, ||ξ(3/2)ⁿ||<2/7 infinitely often. The certificate has depth 15 and 166 blocks across 32 rank strata.
Confinement certificates for powers of rational numbers modulo one
For positive ξ, the fractional parts of ξ(3/2)ⁿ cannot all lie in [1/6,13/24), a specific single interval of length 3/8. A finite block certificate and a general soundness argument establish this exclusion.
Sources
- Kurt Mahler, An unsolved problem on the powers of 3/2
- Artūras Dubickas, On the powers of 3/2 and other rational numbers
- Stephan, confinement-certificate paper
- Stephan, ranked confinement certificates
- Stephan, theorem scope and formal check
- Stephan, verification summary
- Stephan, exact compared statements
- Stephan, prior-art comparison
- Pinned ranked-certificate verification
- Stephan, result and verification summary
- Stephan, formal proof record
- Stephan, compared theorem statement
- Pinned 3/2 certificate verification
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