Let d be the discriminant of an imaginary quadratic field with class number one. The authors give a proof of the well-known conjecture of Gauß that there exist only nine such discriminants. They estimate a linear form in two logarithms of algebraic numbers with integral coefficients and find an upper bound for the absolute value of the discriminant, \(| d| <10^{34}\). They complete the proof for “smaller” discriminants with the help of the theory of continued fractions.
For the history of the conjecture it should be mentioned that the first proof, although incomplete, was given by Heegner at 1952, see also H. M. Stark [Mich. Math. J. 14, 1-27 (1967; Zbl 0148.278)], and C. Meyer [J. Reine Angew. Math. 242, 179-214 (1970; Zbl 0218.12007)]. The first proof which uses estimates of linear forms in logarithms have been obtained by P. Bundschuh and A. Hock [Math. Z. 111, 191-204 (1969; Zbl 0176.332)]. As a consequence of the famous theorem of B. H. Groß and D. B. Zagier [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] one has an effective upper bound for the class number of all imaginary quadratic fields. In class number one case this gives \(| d| <10^{24}\).