Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T18:12:01.396Z Has data issue: false hasContentIssue false
  • English
  • Français

The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms

Published online by Cambridge University Press:  24 October 2008

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
Trinity CollegeCambridge
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper I prove the following results.

Theorem 1. Let f(x, y) = ax2 + bxy + cy2be an indefinite form. Then there exist real (x0, y0) such that

for all (x, y) ≡ (x0, y0) (mod 1). If f represents 0 there are indenumerably many such incongruent (x0, y0).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 1 , January 1952 , pp. 72 - 86
Copyright
Copyright © Cambridge Philosophical Society 1952

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Chabauty, C. and Lutz, E.Sur les approximations linéaires réelles. C.R. Acad. Sci., Paris, 231 (1950), 887–8, 938 –9.Google Scholar
(2)Davenport, H.Indefinite binary quadratic forms and Euclid's algorithm in real quadratic fields. Proc. Lond. math. Soc. (2), 53 (1951), 65–82. (For a proof of the key result but with an unspecified constant instead of 1/128 see Quart. J. Math. (2), 1 (1950), 54–62.)Google Scholar
(3)Davenport, H.Euclid's algorithm in cubic fields of negative discriminant. Acta math., Stockh., 84 (1950), 159–79.CrossRefGoogle Scholar
(4)Davenport, H.Euclid's algorithm in certain quartic fields. Trans. Amer. math. Soc. 68 (1950), 508–32.Google Scholar
(5)Mahler, K.On lattice points in a cylinder. Quart. J. Math. 17 (1946), 16–18.Google Scholar