论文标题
某些整数序列中最不常见倍数的有效估计值
Effective estimates for the least common multiple of some integer sequences
论文作者
论文摘要
该论文致力于研究某些整数序列中最不常见的倍数的估计。我们的研究着重于某些类二次序列的$ \ mathrm {lcm} $的有效界限，以及算术进展和强大的划分序列。首先，我们使用了交换代数的方法和复杂分析的方法来为某些二次序列的$ \ mathrm {lcm} $建立新的非平凡下限。接下来，对强划分性序列的算术特性的更深入的研究使我们获得了这些序列的$ \ mathrm {lcm} $的三个有趣的身份，这些序列概述了Farhi（2009）和Nair（1982）的某些先前身份。结果，我们推导了广义斐波那契序列（所谓的LUCAS序列）的$ \ mathrm {lcm} $的精确估计。我们还开发了一种方法，该方法为Bateman（2002）的渐近结果提供了有效的版本，该结果涉及算术进展的$ \ MATHRM {LCM} $。最后，我们发现可以对后一种方法进行调整以估计序列$（n^2+1）_n $的$ \ mathrm {lcm} $，这尤其允许我们改善Farhi（2005）和Oon（2013）的下限。该论文还包括一些文献结果的一般介绍。
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.
