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Proceedings of the London Mathematical Society Advance Access originally published online on November 23, 2007
Proceedings of the London Mathematical Society 2008 96(2):389-416; doi:10.1112/plms/pdm032
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© 2007 London Mathematical Society

On the representation of integers by quadratic forms

T. D. Browning

School of Mathematics
University of Bristol
Bristol BS8 1TW
United Kingdom

R. Dietmann

Institut für Algebra und Zahlentheorie
Lehrstuhl für Zahlentheorie
Pfaffenwaldring 57
D-70569 Stuttgart
Germany
dietmarr@mathematik.uni-wstuttgart.de

Received 12 December 2006. Revision received 12 April 2007.

Let n ≥ 4 and let Q isin Z[X1, ..., Xn] be a non-singular quadratic form. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q = 0, and when Q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation Q = k is insoluble in integers, despite being soluble modulo every prime power.

2000 Mathematics Subject Classification 11D72 (primary), 11D09, 11P55 (secondary).


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