Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Kane, Daniel M. ; Kominers, Scott Duke

Canadian mathematical bulletin, Tome 57 (2014) no. 3, pp. 551-561

Résumé

For relatively prime positive integers ${{u}_{0}}$ and $r$ , we consider the least common multiple ${{L}_{n}}\,:=\,\text{lcm}\left( {{u}_{0}},\,{{u}_{1}},\,.\,.\,.\,,\,{{u}_{n}} \right)$ of the finite arithmetic progression $\left\{ {{u}_{k}}\,:=\,{{u}_{0}}\,+\,kr \right\}_{k=0}^{n}$ . We derive new lower bounds on ${{L}_{n}}$ that improve upon those obtained previously when either ${{u}_{0}}$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n\,+\,1$ for ${{u}_{0}}$ properly chosen, and is also nearly sharp as $n\,\to \,\infty$ .

Détail

DOI : 10.4153/CMB-2014-017-0

Mots-clés : 11A05, least common multiple, arithmetic progression

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