Dubickas's question on integral parts of powers of square roots settled
The square root of 3 admits a real multiplier making every integral part of its powers even, and among integers only the square root of 2 fails to, answering a question Dubickas posed in 2006 and classifying the whole square-root case.
- Lab
- Independent
- Model
- Claude Fable 5 and Claude Opus 5
- Field
- Mathematics
- Date
- 2026-08-21
- Human collaborators
- Ralf Stephan
- Problem posed
- 2006 · open 20 yrs
What was found
Dubickas splits the reals above 1 into two classes: alpha lies in Z when some nonzero real xi makes every integral part of xi times alpha to the n even, and in S otherwise. At alpha equal to 3/2 this is Mahler's question and still open; his Problem 3 asks which class contains the square root of 3. The answer is that it lies in Z, by an explicit multiplier near 1.3416, and the argument classifies the entire square-root slice: for an integer m at least 2, the square root of m lies in S exactly when m is 2. A further theorem in the paper replaces parity by divisibility by any integer p at least 2.
Novelty check
Dubickas posed the question as Problem 3 of his 2006 Glasgow Mathematical Journal paper (48, 331-336), and it is stated as open there and in the subsequent Z-number literature descending from Mahler. The neighbouring case alpha equal to 3/2 is Mahler's problem and remains open, which fixes the boundary of what is claimed here. No prior resolution of Problem 3 appears. The result is new, and the paper also generalizes past the original question rather than only answering it.
Caveats and known objections
The Lean covers less than the paper. The author states that the case m equal to 3 is what is machine-verified, and the repository flags the thickness computation of section 4.1 and the whole of section 8 as not formalized, so the formal grade attaches to the central claim rather than to every theorem in the write-up. Not peer-reviewed. Autonomy graded ai-led on an unusually explicit disclosure: the author records that the mathematical discovery is the Fable 5 agent's, that the formalization is the Fable 5 and Opus 5 agents', that the agents drafted the prose which the author revised, and that the direction, review and responsibility are the author's. The Lean development carries the same framing in its copyright line.
Machine-checked elsewhere
PalomarPALOMAR-2026-08-23-000002
rwst/Square-Roots at e3184902f139 · checked 2026-08-23
ProvedSZ.sqrtThree_mem_MahlerZSZ.sqrtThree_notMem_SSZ.sqrt_natCast_mem_S_iff
A Lean proof that typechecks against the recorded statement at a pinned commit under a declared axiom set, checked mechanically rather than by human review. It does not certify that a result is new or of research interest: the only filter on that is a language-model screen, and Palomar states that it adds no human editorial step.
Eleven modules, no unfinished proof steps and no declared axioms beyond the three standard ones.
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Entry history (1 event)
- AddedEntered the registry graded Formally verified and AI-led.
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Cite this entry
whataifound.org. (2026). Dubickas's question on integral parts of powers of square roots settled. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-21-dubickas-square-rootsBibTeX
@misc{whataifound-independent-2026-roots,
title = {Dubickas's question on integral parts of powers of square roots settled},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Formally verified. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-08-21-dubickas-square-roots}
}