On the continued fraction expansions of $(1+\protect \sqrt{pq})/2$ and $\protect \sqrt{pq}$

Stéphane R. Louboutin

doi:10.5802/crmath.266

Théorie des nombres

On the continued fraction expansions of

(1 + \sqrt{p q}) / 2

and

\sqrt{p q}

Stéphane R. Louboutin ¹

¹ Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France

Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1201-1205.

Résumé

The evenness and the values modulo $4$ of the lengths of the periods of the continued fraction expansions of $\sqrt{p}$ and $\sqrt{2 p}$ for $p \equiv 3 (mod 4)$ a prime are known. Here we prove similar results for the continued fraction expansion of $\sqrt{p q}$ , where $p, q \equiv 3 (mod 4)$ are distinct primes.

Reçu le : 2021-06-21
Accepté le : 2021-08-27
Publié le : 2021-11-25

DOI : 10.5802/crmath.266

Classification : 11A55, 11R11

Affiliations des auteurs :

Stéphane R. Louboutin ¹

¹ Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {St\'ephane R. Louboutin},
     title = {On the continued fraction expansions of $(1+\protect \sqrt{pq})/2$ and $\protect \sqrt{pq}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1201--1205},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {9},
     year = {2021},
     doi = {10.5802/crmath.266},
     language = {en},
}

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Stéphane R. Louboutin. On the continued fraction expansions of $(1+\protect \sqrt{pq})/2$ and $\protect \sqrt{pq}$. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1201-1205. doi : 10.5802/crmath.266. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.266/

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