Geodesic
Algebraic number fields with large class number
Izvestiya. Mathematics , Tome 8 (1974) no. 5, pp. 967-978.

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We prove that “almost all” real quadratic fields of a given type have a large ideal class number. For example, the number of ideal classes of the fields $\mathbf Q\bigl(\sqrt{m(m+1)(m+2)(m+3)}\,\bigr)$, where $\mathbf Q$ is the field of rational numbers, grows unbounded with $m$, as $m$ ranges through all natural numbers, except for a very sparse sequence. An analogous fact is established for the fields of Ankeny–Brauer–Chowla [5]. 
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V. G. Sprindzhuk. Algebraic number fields with large class number. Izvestiya. Mathematics , Tome 8 (1974) no. 5, pp. 967-978. http://geodesic.mathdoc.fr/item/IM2_1974_8_5_a0/

[1] Gauss K. F., Trudy po teorii chisel, AN SSSR, M., 1959 | MR

[2] Siegel C. L., “Über die Klassenzahl quadratischer Zahlkörper”, Acta Arithmetica, 1 (1935), 83–86 | Zbl

[3] Brauer R., “On the zeta functions of algebraic number fields”, Amer. J. Math., 69 (1947), 243–250 | DOI | MR | Zbl

[4] Landau E., “Abschätzungen von Charaktersummen, Einheiten und Klassenzahlen Nachr.”, Königl. Gesellschaft Wis. Göttingen, 1918, 79–97 | Zbl

[5] Ankeny N. C., Brauer R., Chowla S., “A note on the class-numbers of algebraic number fields”, Amer. J. Math., 78:1 (1956), 51–61 | DOI | MR | Zbl

[6] Sprindzhuk V. G., “Ob otsenkakh edinits algebraicheskikh chislovykh polei”, Dokl. AN BSSR, 15:12 (1971), 1065–1068 | MR

[7] Sprindzhuk V. G., “Ob otsenkakh reshenii uravneniya Tue”, Izv. AN SSSR. Ser. matem., 36 (1972), 712–741

[8] Sprindzhuk V. G., “Svobodnye ot kvadratov deliteli mnogochlenov i chisla klassov idealov algebraicheskikh chislovykh polei”, Acta Arithmetica, 24:2 (1973), 143–149 | Zbl

[9] Siegel C. L., “Abschätzung von Einheiten, Nachr. Acad. Wiss. Göttingen, II”, Math.-Phys. kl., 1969, no. 9, 71–86 | MR | Zbl