Class Number Problem - Gauss's Original Conjectures

Gauss's Original Conjectures

The problems are posed in Gauss's Disquisitiones Arithmeticae of 1801 (Section V, Articles 303 and 304).

Gauss discusses imaginary quadratic fields in Article 303, stating the first two conjectures, and discusses real quadratic fields in Article 304, stating the third conjecture.

Gauss Conjecture (Class number tends to infinity)

Gauss Class Number Problem (Low class number lists)

For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete.

Infinitely many real quadratic fields with class number one

Gauss conjectures that there are infinitely many real quadratic fields with class number one.

The original Gauss class number problem for imaginary quadratic fields is significantly different and easier than the modern statement: he restricted to even discriminants, and allowed non-fundamental discriminants.