Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité
In this note, we first prove that for any strong divisibility sequence $a = \left(a_n\right)_{n\ge 1}$, we have the identity: $\mathrm{lcm}\left\lbrace \binom{n}{0}_{a},\binom{n}{1}_{a},\dots ,\binom{n}{n}_{a}\right\rbrace =\frac{\mathrm{lcm}\left(a_1,\dots ,a_n,a_{n+1}\right)}{a_{n+1}}$ $\left(\for...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2020-07-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.64/ |
| _version_ | 1797651518252384256 |
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| author | Bousla, Sid Ali Farhi, Bakir |
| author_facet | Bousla, Sid Ali Farhi, Bakir |
| author_sort | Bousla, Sid Ali |
| collection | DOAJ |
| description | In this note, we first prove that for any strong divisibility sequence $a = \left(a_n\right)_{n\ge 1}$, we have the identity: $\mathrm{lcm}\left\lbrace \binom{n}{0}_{a},\binom{n}{1}_{a},\dots ,\binom{n}{n}_{a}\right\rbrace =\frac{\mathrm{lcm}\left(a_1,\dots ,a_n,a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n \in \mathbb{N}\right)$, 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 $\left(F_n\right)_n$ the usual Fibonacci sequence, we prove for example that for every positive integer $n$, we have:
\[ \Phi ^{\frac{n^2}{4}-\frac{9}{4}}\le \mathrm{lcm}\left(F_1,\dots ,F_n\right)\le \Phi ^{\frac{n^2}{3}+\frac{4n}{3}}, \]
where $\Phi $ denotes the golden ratio. |
| first_indexed | 2024-03-11T16:16:55Z |
| format | Article |
| id | doaj.art-108a1defe1e24dd5a76614c21e0969b1 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:55Z |
| publishDate | 2020-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-108a1defe1e24dd5a76614c21e0969b12023-10-24T14:19:02ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692020-07-01358448148710.5802/crmath.6410.5802/crmath.64Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilitéBousla, Sid Ali0Farhi, Bakir1Laboratoire de Mathématiques appliquées, Faculté des Sciences Exactes, Université de Bejaia, 06000 Bejaia, AlgérieLaboratoire de Mathématiques appliquées, Faculté des Sciences Exactes, Université de Bejaia, 06000 Bejaia, AlgérieIn this note, we first prove that for any strong divisibility sequence $a = \left(a_n\right)_{n\ge 1}$, we have the identity: $\mathrm{lcm}\left\lbrace \binom{n}{0}_{a},\binom{n}{1}_{a},\dots ,\binom{n}{n}_{a}\right\rbrace =\frac{\mathrm{lcm}\left(a_1,\dots ,a_n,a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n \in \mathbb{N}\right)$, 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 $\left(F_n\right)_n$ the usual Fibonacci sequence, we prove for example that for every positive integer $n$, we have: \[ \Phi ^{\frac{n^2}{4}-\frac{9}{4}}\le \mathrm{lcm}\left(F_1,\dots ,F_n\right)\le \Phi ^{\frac{n^2}{3}+\frac{4n}{3}}, \] where $\Phi $ denotes the golden ratio.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.64/ |
| spellingShingle | Bousla, Sid Ali Farhi, Bakir Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité Comptes Rendus. Mathématique |
| title | Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité |
| title_full | Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité |
| title_fullStr | Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité |
| title_full_unstemmed | Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité |
| title_short | Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité |
| title_sort | identites et estimations concernant le plus petit commun multiple de suites a forte divisibilite |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.64/ |
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