Even floors of powers of quadratic Pisot numbers

For a real quadratic Pisot number α, can a nonzero real ξ make every floor ⌊ξαⁿ⌋ even for n≥1? Dubickas’s set S consists of the numbers for which no such witness exists; the golden ratio is the exceptional candidate among quadratic Pisot numbers.

Why it matters and what counts as progress

Why it matters: The quadratic Pisot classification distinguishes an exceptional number from a whole family of even-floor witnesses.

What counts as progress: Classify all real quadratic Pisot numbers by constructing witnesses or proving their impossibility, including exact formulas for special cases.

How the problem developed

2006 · Open problem [1]
Artūras Dubickas Defined the sets of real numbers whose scaled powers have all-even integral parts and asked whether the complementary set contains any number greater than 2.
21 Aug 2026 · Classification [2][3]
Ralf Stephan with Claude Opus 5 under his direction Classified every quadratic Pisot number: only the golden ratio lies in Dubickas’s exceptional set; all others, including 1 + √2, admit an even-floor witness. The proof is formalized in Lean.

AI contribution

Even integral parts of powers of quadratic Pisot numbers

Ralf Stephan

Among real quadratic Pisot numbers α, the golden ratio is the only one in Dubickas’s set S: every other α admits a nonzero ξ for which all floors ⌊ξαⁿ⌋ are even for n≥1. For α=1+√2, ξ=(2−√2)/4 gives the exact Pell identity ⌊ξ(1+√2)⁽ⁿ⁺¹⁾⌋=Pₙ−[n odd].

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4/Mathlib kernel and Comparator)

Human understandingFew