The problem
Question: determine all imaginary quadratic fields ℚ(√−d) with class number h(−d) = 1. Gauss conjectured the answer is d ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163} (the Heegner numbers).
Gauss's Class Number One Problem
historicHistory & significance
From the Disquisitiones Arithmeticae (§303). Heegner submitted a proof in 1952 using modular functions and Weber invariants; it was dismissed as containing gaps (which were cosmetic, not essential) and he died before vindication. Stark (1967) and Baker (1971) gave independent proofs via different methods; Birch–Chowla–Schinzel and Deuring confirmed Heegner was essentially correct.
Connected problems
- related The Stark Conjectures
More in algebraic number theory
- same area The Stark Conjectures
References
- primaryRead the primary source →
- sourceWikipedia — Heegner number
The resolution (human proof)
Resolved 1967 by Harold Stark and Alan Baker (independently), confirming Kurt Heegner's 1952 approach. Weinberger (1973) extended the method to rule out the last GRH-conditional exceptional d. Goldfeld–Gross–Zagier (1983-86) later provided an EFFECTIVE algorithm for all class numbers, solving the companion problem for h(−d) = 2 as well. The Heegner numbers' connection to j-invariants and modular functions makes this a bridge between algebraic number theory and complex multiplication theory — and a cautionary tale about trusting peer review over computation.
How to check it: the answer is the nine Heegner numbers d ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163} — verifiable today by genus theory plus finite computation. Heegner's 1952 argument had a gap the referees missed; Stark and Baker independently found correct proofs in 1966–67, which is why the theorem bears three names and the cautionary tale.
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