, Volume 70, Issue 6, pp 2059-2076

Quadratic irrationals with fixed period length in the continued fraction expansion

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We present an algorithm that makes it possible to write out all quadratic irrationals of the form \(\sqrt D \) , that have a given even period length in the continued fraction expansion. It turns out that in the expansion $$\sqrt D = \left[ {b_0 ,\overline {l_1 ,...,l_L ,...,l_1 ,2b_0 } } \right]$$ λ={l1, ..., lL+1} is almost arbitrary, and b0 (and, consequently D) runs through a very narrow sequence depending on λ. We obtain a summation formula for the class numbers of indefinite binary forms with discriminant D with D≤X for which the set λ is fixed.

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 5–30, 1991.