Length of the period of a quadratic irrational

E. V. Podsypanin

Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 5, Tome 82 (1979), pp. 95-99

Résumé

Let $\xi$ be a real quadratic irrational of discriminant $D=f^2D_1>0$, where $D_1$ is the fundamental discriminant of the field $\mathbf Q(\sqrt{D})$, $\chi(n)$ and $h$ are the character and the number of classes of the field $\mathbf Q(\sqrt{D})$, $L(1,\chi)=\sum^\infty_{n=1}\frac{\chi(n)}{n}$, respectively, and $$ l<\frac{\omega}{\log\dfrac{1+\sqrt{5}}{2}}\cdot\frac{D^{\frac12}L(1,\chi)}{h}, $$ proves the following estimate for the length $l$ of the period of the expansion of $\xi$ into a continued fraction: where $\omega=1$ if $f=1$ and $\omega=2$ 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 $f=1$, $D_1\equiv0$ $(\operatorname{mod}4)$.