2019/07/15 by Sid Ali Bousla, Bousla, Sid Ali, Bakir Farhi +1
Computer Science · Mathematics · Physics and Astronomy · #11B39 #11B83 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Primary 11A05 #Secondary 11B65
paper · pdf · doi:10.48550/arxiv.1907.06700
openalex publication_date 2019/07/15 · openalex created_date 2019/07/23 · openalex updated_date 2026/07/28
In this paper, we first prove that for any strong divisibility sequences \boldsymbola = (an)n≥ 1, we have the identity: lcm \lbrace \binomn0_\bfa, \binomn1_\bfa,…, \binomnn_\bfa \rbrace = \fraclcm (a1,… , an , an+1)an+1 (∀ n ≥ 1), generalizing the identity of Farhi (obtained in 2009 for an=n). Then, we derive from this one some other interesting identities. Finally, we apply those identities to estimate the least common multiple of the consecutive terms of some Lucas sequences. Denoting by (Fn)n the usual Fibonacci sequence, we prove for example that for all n ≥ 1, we have Φ(n2)/(4)-(9)/(4) ≤ lcm(F1,…,Fn) ≤ Φ(n2)/(3)+(4n)/(3) , where Φ denotes the golden ratio.