On the period of the continued fraction expansion for $\sqrt{d}$
- Authors: Korolev M.A.1
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Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
- Issue: Vol 89, No 1 (2025)
- Pages: 30-53
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/303936
- DOI: https://doi.org/10.4213/im9587
- ID: 303936
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Abstract
If $d$ is not a perfect square, we define $T(d)$ as the length of theminimal period of the simple continued fraction expansion for $\sqrt{d}$.Otherwise, we put $T(d)=0$. In the recent paper (2024), F. Battistoni,L. Grenie and G. Molteni established (in particular) an upper boundfor the second moment of $T(d)$ over the segment $x
About the authors
Maxim Aleksandrovich Korolev
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Author for correspondence.
Email: korolevma@mi-ras.ru
Doctor of physico-mathematical sciences, no status
References
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- F. Battistoni, L. Grenie, G. Molteni, “The first and second moment for the length of the period of the continued fraction expansion for $sqrt{d}$”, Mathematika, 70:4 (2024), e12273, 12 pp.
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- Н. М. Коробов, Тригонометрические суммы и их приложения, Наука, М., 1989, 240 с.
- S. Bettin, V. Chandee, “Trilinear forms with Kloosterman fractions”, Adv. Math., 328 (2018), 1234–1262
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