(for negative integers d) having class number n. It is named after Carl Friedrich Gauss.
The problems are posed in Gauss's Disquisitiones Arithmeticae of 1801 (Section V, Articles 303 and 304).
[1] 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.
[4] The complete list of imaginary quadratic fields with class number 1 is
where d is one of The general case awaited the discovery of Dorian Goldfeld in 1976 that the class number problem could be connected to the L-functions of elliptic curves.
[5] With the proof of the Gross–Zagier theorem in 1986, a complete list of imaginary quadratic fields with a given class number could be specified by a finite calculation.
It may well be the case that class number 1 for real quadratic fields occurs infinitely often.
The Cohen–Lenstra heuristics[6] are a set of more precise conjectures about the structure of class groups of quadratic fields.