On primitively universal quadratic forms
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In 1999, Manjul Bhargava proved the Fifteen Theorem and showed that there are exactly 204 universal positive definite integral quaternary quadratic forms. We consider primitive representations of quadratic forms and investigate a primitive counterpart to the Fifteen Theorem. In particular, we give an efficient method for deciding whether a positive definite integral quadratic form in four or more variables with odd square-free determinant is almost primitively universal.
Keywordsalmost primitively universal quadratic forms p-adic symbols
MSC11E20 11E25 11E08 11E99
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- 1.M. Bhargava, Finiteness theorems for quadratic forms, preprint.Google Scholar
- 2.M. Bhargava, On the Conway–Schneeberger Fifteen Theorem, in E. Bayer-Fluckiger, D. Lewis, and A. Ranicki (Eds.), Proceedings of the Conference on Quadratic Forms and Their Applications, AMS Bookstore, Dublin, 1999, pp. 27–38.Google Scholar
- 3.J. Borhnak and B.-K. Oh, Almost universal quadratic forms: An effective solution of a problem of Ramanujan, preprint.Google Scholar
- 4.J.W.S. Cassels, Rational Quadratic Forms, London Math. Soc. Monographs, Vol. 13, Academic Press, London, New York, 1978.Google Scholar
- 5.J.H. Conway, Universal quadratic forms and the Fifteen Theorem, in E. Bayer-Fluckiger, D. Lewis, and A. Ranicki (Eds.), Proceedings of the Conference on Quadratic Forms and Their Applications, AMS Bookstore, Dublin, 1999, pp. 23–26.Google Scholar
- 7.J.H. Conway and J.A. Sloane, Sphere Packings, Lattices and Groups, Grundlehren Math. Wiss, Vol. 290, Springer-Verlag, New York, Berlin, 1988.Google Scholar