Comptes Rendus
Number Theory
Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité
[Identities and estimations involving the least common multiple of strong divisibility sequences]
Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 481-487.

In this note, we first prove that for any strong divisibility sequence a=a n n1 , we have the identity: lcmn 0 a ,n 1 a ,,n n a =lcma 1 ,,a n ,a n+1 a n+1 n, generalizing the identity of Farhi (obtained in 2009 for a n =n). Then, we derive from it other interesting identities. Finally, we apply those identities to estimate the least common multiple of the consecutive terms of some Lucas sequences. Denoting by F n n the usual Fibonacci sequence, we prove for example that for every positive integer n, we have:

Φ n 2 4-9 4 lcmF 1 ,,F n Φ n 2 3+4n 3 ,

where Φ denotes the golden ratio.

Dans cette note, nous montrons d’abord que pour toute suite à forte divisibilité a=a n n1 , on a l’identité : ppcmn 0 a ,n 1 a ,,n n a =ppcma 1 ,,a n ,a n+1 a n+1 n, généralisant l’identité de Farhi (obtenue en 2009 pour a n =n). Par suite, nous en déduisons quelques autres identités intéressantes. Finalement, nous appliquons ces identités pour estimer le plus petit commun multiple des termes consécutifs de certaines suites de Lucas. En désignant par F n n la suite de Fibonacci usuelle, nous montrons par exemple que pour tout entier n1, on a :

Φ n 2 4-9 4 ppcmF 1 ,,F n Φ n 2 3+4n 3 ,

Φ désigne le nombre d’or.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.64
Sid Ali Bousla 1; Bakir Farhi 1

1 Laboratoire de Mathématiques appliquées, Faculté des Sciences Exactes, Université de Bejaia, 06000 Bejaia, Algérie
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     author = {Sid Ali Bousla and Bakir Farhi},
     title = {Identit\'es et estimations concernant le plus petit commun multiple de suites \`a forte divisibilit\'e},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {481--487},
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     year = {2020},
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Sid Ali Bousla; Bakir Farhi. Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 481-487. doi : 10.5802/crmath.64. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.64/

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