Gauss’s conjecture on infinitely many real quadratic fields of class number one

Establish the existence of infinitely many real quadratic number fields Q(√D) with class number h(D) = 1; equivalently, show that the set of positive fundamental discriminants D for which the real quadratic field Q(√D) has class number one is infinite.

Background

The paper studies class numbers of specific families of real quadratic fields and highlights broader unresolved questions in the area. In this context, the authors recall Gauss’s famous conjecture concerning the infinitude of real quadratic fields with class number one, emphasizing that class numbers remain mysterious despite substantial progress.

Their results focus on proving lower bounds and class number behavior for explicit discriminant families, which contrasts with Gauss’s conjecture that seeks infinitely many instances where the class number is exactly one. This places the conjecture as a motivating open problem within the paper’s thematic scope.