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Small solutions of quadratic and quartic congruences

Published online by Cambridge University Press:  26 February 2010

R. C. Baker
R. C. Baker
Affiliation:
Royal Holloway College, Egham, Surrey
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Extract

The present investigation was suggested by a theorem of A. Schinzel, H. P. Schlickewei and W. M. Schmidt [6]: let Q(x) = Q(x1,…, xs) be a quadratic form with integer coefficients. Then for each natural number m there are integers x1,…, xs satisfying

and having

Type
Research Article
Information
Mathematika , Volume 27 , Issue 1 , June 1980 , pp. 30 - 45
Copyright
Copyright © University College London 1980

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References

1.Baker, R. C. and Schmidt, W. M.. “Diophantine problems with variables restricted to 0 and 1”, J. Number Theory (to appear).Google Scholar
2.Birch, B. J.. “Homogeneous forms of odd degree in a large number of variables”, Mathematika, 4 (1957), 102–105.CrossRefGoogle Scholar
3.Borevich, Z. I. and Shafarevich, I. R.. Number Theory (Academic Press, New York, 1966).Google Scholar
4.Cassels, J. W. S.. “Bounds for the least solutions of homogeneous quadratic equations”, Proc. Cambridge Philos. Soc, 51 (1955), 262–264.CrossRefGoogle Scholar
5.Cassels, J. W. S.. An introduction to Diophantine approximation (Cambridge University Press, London, 1957).Google Scholar
6.Schinzel, A., Schlickewei, H. P. and Schmidt, W. M.. “Small solutions of quadratic congruences and small fractional parts of quadratic forms”, Acta Arith. (to appear).Google Scholar
7.Schmidt, W. M.. “Small zeros of additive forms in many variables”, Trans. Amer. Math. Soc, 248 (1979), 121–133.CrossRefGoogle Scholar
8.Schmidt, W. M.. “Diophantine inequalities for forms of odd degree”, Advances in Math, (to appear).Google Scholar