Geodesic
On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials $f$ over algebraic number fields
Sbornik. Mathematics, Tome 213 (2022) no. 3, pp. 412-442 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain a complete description of the fields $\mathbb K$ that are extensions of $\mathbb Q$ of degree at most $3$ and the cubic polynomials $f \in\mathbb K[x]$ such that the expansion of $\sqrt{f}$ into a continued fraction in the field of formal power series $\mathbb K((x))$ is periodic. We prove a finiteness theorem for cubic polynomials $f \in\mathbb K[x]$ with a periodic expansion of $\sqrt{f}$ for extensions of $\mathbb Q$ of degree at most $6$. We obtain a description of the periodic elements $\sqrt{f}$ for the cubic polynomials $f(x)$ defining elliptic curves with points of order $3 \le N\le 42$, $N \ne 37, 41$. Bibliography: 19 titles.
Keywords: elliptic field, $S$-units, continued fractions, periodicity
Mots-clés : torsion points. 
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V. P. Platonov; V. S. Zhgoon; M. M. Petrunin. On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials $f$ over algebraic number fields. Sbornik. Mathematics, Tome 213 (2022) no. 3, pp. 412-442. http://geodesic.mathdoc.fr/item/SM_2022_213_3_a6/

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