Abstract
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
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.
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References
B. A. Venkov, Elementary Number Theory [in Russian], Moscow-Leningrad (1937).
A. S. Pen and B. F. Skubenko, “Upper bound for the period of a quadratic irrational,” Mat. Zametki,5, No. 4, 413–417 (1969).
E. P. Golubeva, “The length of the period of a quadratic irrational,” Mat. Sb.,123, No. 1, 120–129 (1984).
C. D. Patterson and H. C. Williams, “Some periodic continued fractions with long periods,” Math. Comput.,44, No. 170, 523–532 (1985).
P. C. Sarnak, “Class number of indefinite binary quadratic forms. I, II,” J. Number Theory,15, No. 2, 229–247 (1982);21, No. 3, 333–346 (1985).
C. Hooley, “On the Pellian equation and the class number of indefinite binary quadratic forms,” J. Reine und Angew. Math.,353, 98–131 (1984).
S. Chowla and J. Friedlander, “Class number and quadratic residues,” Glasgow Math. J.,17, 47–52 (1976).
R. A. Mollin, “Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla,” Proc. Am. Math. Soc.,122, No. 1, 17–21 (1988).
V. G. Sprindzhuk, Classical Diophantine Equations in Two Unknowns [in Russian], Moscow-Leningrad (1982).
E. P. Golubeva, “Period lengths for continued fraction expansions of quadratic irrationals and the class numbers of real quadratic fields. II,” Zap. Nauchn. Semin. LOMI,168, 11–22 (1988).
Y. Yamamoto, “Real quadratic number fields with large fundamental units,” Osaka J. Math.,8, No. 2, 261–270 (1971).
E. P. Golubeva, “Period lengths of continued fraction expansions of quadratic irrationals and class numbers of real quadratic fields. I,” Zap. Nauchn. Sem. LOMI,160, 72–81 (1987).
E. P. Golubeva, “Indefinite binary quadratic forms with large class numbers,” Zap. Nauchn. Semin. LOMI,185, 13–21 (1990).
E. P. Golubeva, “On a problem of Waring for ternary quadratic forms and arbitrary even degree,” Zap. Nauchn. Semin. LOMI,144, 27–37 (1985).
O. M. Fomenko, “Estimating scalar Peterson products for parabolic forms and arithmetic applications,” Zap. Nauchn. Semin. LOMI,168, 158–179 (1988).
E. V. Podsypanin, “Number of integer points in an elliptic region (remark on a theorem of A. V. Malyshev),” Zap. Nauchn. Semin. LOMI,82, 100–102 (1979).
H. Montgomery, “Topics in multiplicative number theory,” Lect. Notes Math.,227 (1971).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 5–30, 1991.
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Golubeva, E.P. Quadratic irrationals with fixed period length in the continued fraction expansion. J Math Sci 70, 2059–2076 (1994). https://doi.org/10.1007/BF02111323
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DOI: https://doi.org/10.1007/BF02111323