, Volume 18, Issue 6, pp 919-923

Length of the period of a quadratic irrational

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Abstract

Let ξ be a real quadratic irrational of discriminant D=f2Di>0, where Di 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, D1≡0 (mod4).

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 95–99, 1979.