Identités et estimations concernant le plus petit commun multiple de suites à forte divisibilité

Bousla, Sid Ali; Farhi, Bakir

Comptes Rendus. Mathématique, Tome 358 (2020) no. 4, pp. 481-487.

Dans cette note, nous montrons d’abord que pour toute suite à forte divisibilité $a={\left(a_n\right)}_{n\ge 1}$, on a l’identité : $\mathrm{ppcm}\left\{{\left(\genfrac{}{}{0pt}{}{n}{0}\right)}_a,{\left(\genfrac{}{}{0pt}{}{n}{1}\right)}_a,\cdots ,{\left(\genfrac{}{}{0pt}{}{n}{n}\right)}_a\right\}=\frac{\mathrm{ppcm}\left(a_1,\cdots ,a_n,a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n\in \mathbb{N}\right)$, 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 ${\left(F_n\right)}_n$ la suite de Fibonacci usuelle, nous montrons par exemple que pour tout entier $n\ge 1$, on a :

$${\Phi }^{\frac{n^2}{4}-\frac{9}{4}}\le \mathrm{ppcm}\left(F_1,\cdots ,F_n\right)\le {\Phi }^{\frac{n^2}{3}+\frac{4n}{3}},$$

où $\Phi$ désigne le nombre d’or.

In this note, we first prove that for any strong divisibility sequence $a={\left(a_n\right)}_{n\ge 1}$, we have the identity: $\mathrm{lcm}\left\{{\left(\genfrac{}{}{0pt}{}{n}{0}\right)}_a,{\left(\genfrac{}{}{0pt}{}{n}{1}\right)}_a,\cdots ,{\left(\genfrac{}{}{0pt}{}{n}{n}\right)}_a\right\}=\frac{\mathrm{lcm}\left(a_1,\cdots ,a_n,a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n\in \mathbb{N}\right)$, 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:

$${\Phi }^{\frac{n^2}{4}-\frac{9}{4}}\le \mathrm{lcm}\left(F_1,\cdots ,F_n\right)\le {\Phi }^{\frac{n^2}{3}+\frac{4n}{3}},$$

where $\Phi$ denotes the golden ratio.

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