[Identities and estimations involving the least common multiple of strong divisibility sequences]
In this note, we first prove that for any strong divisibility sequence , we have the identity: , generalizing the identity of Farhi (obtained in 2009 for ). 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 the usual Fibonacci sequence, we prove for example that for every positive integer , we have:
where denotes the golden ratio.
Dans cette note, nous montrons d’abord que pour toute suite à forte divisibilité , on a l’identité : , généralisant l’identité de Farhi (obtenue en 2009 pour ). 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 la suite de Fibonacci usuelle, nous montrons par exemple que pour tout entier , on a :
où désigne le nombre d’or.
Revised:
Accepted:
Published online:
Sid Ali Bousla 1; Bakir Farhi 1
@article{CRMATH_2020__358_4_481_0, 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}, publisher = {Acad\'emie des sciences, Paris}, volume = {358}, number = {4}, year = {2020}, doi = {10.5802/crmath.64}, language = {fr}, }
TY - JOUR AU - Sid Ali Bousla AU - Bakir Farhi TI - Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité JO - Comptes Rendus. Mathématique PY - 2020 SP - 481 EP - 487 VL - 358 IS - 4 PB - Académie des sciences, Paris DO - 10.5802/crmath.64 LA - fr ID - CRMATH_2020__358_4_481_0 ER -
%0 Journal Article %A Sid Ali Bousla %A Bakir Farhi %T Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité %J Comptes Rendus. Mathématique %D 2020 %P 481-487 %V 358 %N 4 %I Académie des sciences, Paris %R 10.5802/crmath.64 %G fr %F CRMATH_2020__358_4_481_0
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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