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Solutions of Diophantine equations and divisibility of class numbers of complex quadratic fields

Published online by Cambridge University Press:  18 May 2009

R. A. Mollin
R. A. Mollin
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4, Canadaramollin@math.ucalgary.ca
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We show how the solution to certain diophantine equations involving the discriminant of complex quadratic fields leads to the divisibility of the class numbers of the underlying fields. This not only generalizes certain results in the literature such as [2], [4]–[6] but also shows why certain hypotheses made in these results are actually unnecessary since, as our criteria demonstrate, these hypotheses are forced by the solution of the diophantine equations involved. Our methods are based only on the most elementary properties of a principal ideal in a complex quadratic field.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 38 , Issue 2 , May 1996 , pp. 196 - 197
Copyright
Copyright © Glasgow Mathematical Journal Trust 1996

References

REFERENCES

1.Alter, R. and Kubota, K. K., The diophantine equation x2 + 11 = 3n and a related sequence, J. Number Theory 7 (1975), 5–10.CrossRefGoogle Scholar
2.Ankeny, N. C. and Chowla, S., On the divisibility of the class number of quadratic fields, Pacific J. Math. 5 (1955), 321–324.CrossRefGoogle Scholar
3.Cohn, J. H. E., The Diophantine Equation x2 +3 = yn, Glasgow Math. J. 35 (1993), 203–206.CrossRefGoogle Scholar
4.Cowles, M. J., On the divisibility of the class number of imaginary quadratic fields, j. Number Theory 12 (1980), 113–115.CrossRefGoogle Scholar
5.Gross, B. H. and Rohrlich, D. E., Some results on the Mordell–Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978), 201–224.CrossRefGoogle Scholar
6.Mollin, R. A., Diophantine equations and class numbers, J. Number Theory 24 (1986), 7–19.CrossRefGoogle Scholar
7.Mollin, R. A., Orders in quadratic fields III, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), 176–181.CrossRefGoogle Scholar