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On the minimum of zero indefinite binary quadratic forms

Published online by Cambridge University Press:  26 February 2010

Mary E. Gbur
Mary E. Gbur
Affiliation:
Texas A & M University, Department of Mathematics, College Station, Texas 77843, U.S.A.
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Abstract

We consider a Markoff spectrum for the set of indefinite binary quadratic forms with real coefficients which represent zero non-trivially. As was done for the classical Markoff spectrum, we show that 1/3 is the largest accumulation point of the set and explicitly determine the countably infinite number of elements greater than 1/3. Unlike the situation for the classical Markoff spectrum, there is a countably infinite number of limit points greater than 1/3.

Type
Research Article
Information
Mathematika , Volume 25 , Issue 1 , June 1978 , pp. 94 - 106
Copyright
Copyright © University College London 1978

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References

1.Dickson, L. E.. Studies in the theory of numbers (The University of Chicago Press, 1930), Chapter VII.Google Scholar
2.Gurwood, C.. Diophantine approximation and the Markov Chain, Ph.D. dissertation (New York University, 1976).Google Scholar
3.Jackson, T. H.. “On the minima of zero binary quadratic forms”, J. London Math. Soc. (2), 14 (1976), 178–182.CrossRefGoogle Scholar
4.Markoff, A.. “Sur les formes quadratiques binaires indefinies”, Math. Ann., 15 (1879), 381–409; 17 (1880), 379–399.CrossRefGoogle Scholar
5.Perron, O.. “Über die Approximation irrationaler Zahlen durch rationale, I-II”, S.-B. Heidelberger Akad. Wiss. Abh. 4 (1921), 3–17; Abh. 8 (1921), 1–21.Google Scholar
6.Perron, O.. Die Lehre von den Kettenbrüche, Band 1 (B. G. Teubner, Stuttgart, 1954).Google Scholar