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Kaplansky's problem on the values of the $u$-invariant: u_invariant_values

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Statement

The values of the uu-invariant, stated as in [MerkurjevParimala2025, §5.1]: which integers are uu-invariants of fields (of characteristic not 22)? The question goes back to [Kaplansky1953], who conjectured that only powers of 22 occur (u_invariant_values.variants.kaplansky_conjecture). Every positive even integer is a uu-invariant (u_invariant_values.variants.even), 33, 55 and 77 are not (u_invariant_values.variants.not_three, u_invariant_values.variants.not_five, u_invariant_values.variants.not_seven), and it is expected that every odd integer ≥9\ge 9 is (u_invariant_values.variants.odd). Granting the preprint [Karpenko2026] (u_invariant_values.variants.karpenko), the answer is known for every nn except n=2r−1n = 2^r - 1 and n=2r−3n = 2^r - 3 with r≥4r \ge 4.

Lean statement

theorem u_invariant_values :
    ∀ n, n ∈ u_invariant_values.answer ↔ IsUInvariant n := by
  sorry

Sources(1)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/Paper/KaplanskyUInvariant.lean.

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