Enumeration of imaginary quadratic fields with bounded class number
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Which imaginary quadratic fields have class number at most a given bound? A complete enumeration requires both a list of fields and a proof that none outside the searched discriminant range are missing. The recorded target raises the class-number cutoff to 2000 without assuming the generalized Riemann hypothesis.
Why it matters and what counts as progress
Why it matters: Complete class-number lists classify these arithmetic fields and distinguish unconditional results from enumerations that depend on an unproved hypothesis.
What counts as progress: Prove explicit discriminant bounds, extend a complete enumeration to a larger class-number cutoff, or remove a conditional assumption. The analytic bound and the sieve, tables and extraction used for the final list must all support completeness; checking the bound alone does not verify the enumeration.
How the problem developed
- 2004 · Classification [1]
Mark Watkins Completed the classification of imaginary quadratic fields with class number at most 100, using theory and a substantial computation.
- 1 Oct 2026 · Reported claim [2]
OpenAI internal model Released an explicit argument excluding Landau–Siegel zeros, which Kimura later used as an input to bound discriminants for the class-number enumeration.
- 15 Oct 2026 · Classification [3]
Iwao Kimura, with Claude Opus 5.5 under his direction Determined all 16,453,493 imaginary quadratic fields with class number at most 2000. The paper formalizes the zero-gap and discriminant bounds, while the sieve and table extraction remain computational.
AI contribution
Claude-assisted work extends unconditional imaginary quadratic class-number lists to 2000
Kimura reports exactly 16,453,493 imaginary quadratic fields with class number at most 2000; the largest absolute field discriminant on the list is 1,062,350,083.
Sources
Website timeline
Publication dates and editorial changes.
Claude-assisted work extends unconditional imaginary quadratic class-number lists to 2000