Square roots in Dubickas’s even-floor power set
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For α>1, does some nonzero real ξ make ⌊ξαⁿ⌋ even for every n≥1? Let Z be the set of α admitting such a witness; the square-root question asks which √m belong to Z for integers m≥2.
Why it matters and what counts as progress
Why it matters: The integer square-root case makes Dubickas’s floor-parity existence question an explicit arithmetic classification.
What counts as progress: Construct a witness for √3 and other square roots or prove an exact exceptional classification.
How the problem developed
- 2006 · Definition and open question [1]
Artūras Dubickas Defined the set of bases α for which some nonzero ξ makes every floor ⌊ξαⁿ⌋ even, and reviewed Mahler’s related unresolved question about 3/2.
- August 2026 · Theorem claim [2]
Ralf Stephan Reported that √m belongs to Dubickas’s even-floor set for every integer m≥2 except m=2, including an explicit witness for √3.
AI contribution
Even integral parts of powers of square roots
For Z={α>1: some nonzero real ξ makes every floor ⌊ξαⁿ⌋ even for n≥1}, Stephan constructs a witness showing √3∈Z and proves that among integer square roots √m with m≥2, only √2 lies outside Z.
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AI-discovered construction answers Dubickas’s square-root question