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A Note on Cubic Equations

Published online by Cambridge University Press:  24 October 2008

James Fowler
James Fowler
Affiliation:
Trinity CollegeCambridge
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Extract

It was proved by Davenport (2) that a cubic form in n variables with rational coefficients always represents zero if n > 32; and this condition was later (3) relaxed to n ≥ 29. The object of the present paper is to establish conditions under which a cubic form will represent every rational number other than zero, for rational values of the variables. The main result is as follows.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 2 , April 1962 , pp. 165 - 169
Copyright
Copyright © Cambridge Philosophical Society 1962

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References

REFERENCES

(1)Birch, B. J.Forms in many variables. Proc. Roy. Soc. London. Ser. A, (to appear).Google Scholar
(2)Davenport, H.Cubic forms in 32 variables. Philos. Trans. Roy. Soc. London. Ser. A, 251 (1959), 193–232.Google Scholar
(3)Davenport, H. Cubic forms in 29 variables. Proc. Roy. Soc. London. Ser. A, (to appear).Google Scholar
(4)Segre, B.A parametric solution of the indeterminate cubic equation z 2 = f(x, y). J. London Math. Soc. 19 (1943), 226–232.CrossRefGoogle Scholar
(5)Watson, G. L.Cubic forms representing arithmetic progressions. Proc. Cambridge Philos. Soc. 55 (1959), 270–273.CrossRefGoogle Scholar
(6)Whitehead, R. F.A rational parametric solution of the cubic indeterminate equation z 2 = f(x, y). J. London Math. Soc. 19 (1944), 68–71.CrossRefGoogle Scholar