# Length of the period of a quadratic irrational

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## Abstract

Let ξ be a real quadratic irrational of discriminant D=f

^{2}D_{i}>0, where D_{i}is the fundamental discriminant of the field and h are the character and the number of classes of the field, respectively, and\(L\left( {1,\chi } \right) = \sum\limits_{n = 1}^\infty {\frac{{\chi \left( n \right)}}{n}} \) proves the following estimate for the length l of the period of the expansion of ξ into a continued fraction: where ω=1 if f=1 and ω=2 if f>1. A. S. Pen and B. F. Skubenko (Mat. Zametki, 5, No. 4, 413–482 (1969)) have proved this estimate in the case f=1, D_{1}≡0 (mod4).## Preview

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### Literature cited

- 1.Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York (1966).Google Scholar
- 2.L. V. Danilov and G. V. Danilov, “An estimate from above for the period of quadratic irrational,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat.,19, No. 9, 19–24 (1975).Google Scholar
- 3.E. A. Murzaev, “On an estimate of the upper bound of the length of the period of a continued fraction,” Volzhskii Mat. Sb., No. 5, 260–261 (1966).Google Scholar
- 4.A. S. Pen and B. F. Skubenko, “An upper bound for the period of a quadratic irrationality,” Mat. Zametki,5, No. 4, 413–418 (1969).Google Scholar
- 5.E. V. Podsypanin, “An estimate from above for the length of the period of a quadratic irrationality,” in: Tezisy Dok. i Soobshch. Vsesoyuz. Shk. po Teorii Chisel, Dushanbe (1977), p. 100.Google Scholar
- 6.J. H. E. Cohn, “The length of the period of the simple continued fraction of\(\sqrt d \),” Pac. J. Math.,71, No. 1, 21–32 (1977).Google Scholar
- 7.D. R. Hickerson, “Length of period of simple continued fraction expansion of,” Pac. J. Math.,46, 429–432 (1973).Google Scholar
- 8.K. E. Hirst, “The length of periodic continued fractions,” Monatsh. Math.,76, 428–435 (1972).Google Scholar
- 9.R. C. Stanton, C. Sudler, Jr., and H. C. Williams, “An upper bound for the period of the simple continued fraction for,” Pac. J. Math.,67, No. 2, 525–536 (1976).Google Scholar

## Copyright information

© Plenum Publishing Corporation 1982