Effective estimates for the least common multiple of some integer sequences

TL;DR

This thesis develops new effective bounds and identities for the least common multiple of various integer sequences, including quadratic, arithmetic, and strong divisibility sequences, improving existing estimates and generalizing previous results.

Contribution

It introduces novel bounds and identities for the LCM of quadratic, arithmetic, and divisibility sequences, extending prior work and providing effective estimates for specific sequences.

Findings

New nontrivial lower bounds for quadratic sequences' LCM

Identities relating to the LCM of strong divisibility sequences

Improved lower bounds for the LCM of sequences like n^2+1

Abstract

This thesis is devoted to studying estimates of the least common multiple of some integer sequences. Our study focuses on effective bounding of the $\mathrm{lcm}$ of some class of quadratic sequences, as well as arithmetic progressions and strong divisibility sequences. First, we have used methods of commutative algebra and complex analysis to establish new nontrivial lower bounds for the $\mathrm{lcm}$ of some quadratic sequences. Next, a more profound study of the arithmetic properties of strong divisibility sequences allowed us to obtain three interesting identities involving the $\mathrm{lcm}$ of these sequences, which generalizes some previous identities of Farhi (2009) and Nair (1982); as consequences, we have deduced a precise estimates for the $\mathrm{lcm}$ of generalized Fibonacci sequence (the so-called Lucas sequences). We have also developed a method that provides an effective version to the asymptotic result of Bateman (2002) concerning the $\mathrm{lcm}$ of an arithmetic progression. Finally, we found that the latter method can be adapted to estimate the $\mathrm{lcm}$ of the sequence $(n^2+1)_n$, which allowed us in particular to improve the lower bounds of Farhi (2005) and Oon (2013). The thesis also includes a general presentation of some literature results.