Math Conjectures

Real Quadratic Class-Number-One Conjecture

OPENLandmarkConjectureProposed c. 1801 · Full conjecture

Canonical statement

There are infinitely many positive squarefree integers dd such that the ordinary ideal class number satisfies
h(Q(d))=1. h\bigl(\mathbb Q(\sqrt d)\bigr)=1.
Equivalently, the ring of integers of Q(d)\mathbb Q(\sqrt d) is a principal ideal domain.
View source LaTeX
There are infinitely many positive squarefree integers \(d\) such that the ordinary ideal class number satisfies
\[
  h\bigl(\mathbb Q(\sqrt d)\bigr)=1.
\]
Equivalently, the ring of integers of \(\mathbb Q(\sqrt d)\) is a principal ideal domain.
NT-QUAD-043:[MollinWilliams1988PrimeValues][MollinWilliams1990ClassNumber][CohenLenstra1984ClassGroups]
Number theory

Notes

The real quadratic class-number-one conjecture asks for infinitely many positive squarefree dd for which the ordinary ideal class group of Q(d)\mathbb Q(\sqrt d) is trivial. This is equivalent to its ring of integers being a principal ideal domain. The prediction is traditionally traced to Gauss's study of binary quadratic forms; later accounts place it among the basic unresolved class-number problems for real quadratic fields [MollinWilliams1990ClassNumber].

Many examples and parameterized candidate families are known, and prime-producing criteria illuminate special cases [MollinWilliams1988PrimeValues]. Cohen–Lenstra heuristics predict that trivial and small class groups should remain abundant [CohenLenstra1984ClassGroups], but the heuristic does not prove even infinitude. In contrast with imaginary quadratic fields, the analytic class-number formula here contains the regulator, governed by a large and erratic fundamental unit.

No unconditional method controls that regulator well enough across an infinite family to force class number one. The card uses the ordinary class number: the narrow class number is different when no unit of norm −1-1 exists and would define a distinct, stronger distributional question.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.