Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
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DOI：
10.4153/cmb-2014-017-0
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发表时间：
2010-12
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影响因子：
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通讯作者：
D. Kane;Scott Duke Kominers
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文献类型：
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作者：
D. Kane;Scott Duke Kominers
摘要 对于互质正整数 ${{u}_{0}}$ 和 $r$ ，我们考虑有限算术级数 $\left\{ 的最小公倍数 ${{L}_{n}}\,:=\,\text{lcm}\left( {{u}_{0}},\,{{u}_{1}},\,.\,.\,.\,,\,{{u}_{n}} \right)$ {{u}_{k}}\,:=\,{{u}_{0}}\,+\,kr \right\}_{k=0}^{n}$ 。我们推导出 ${{L}_{n}}$ 的新下界，该下界改进了之前在 ${{u}_{0}}$ 或 $n$ 较大时获得的下界。当 $r$ 为质数时，对于正确选择的 ${{u}_{0}}$，我们的最佳界限急剧上升至 $n\,+\,1$ 因子，并且也接近 $n\,\to \,\infty$ 急剧上升。
Abstract 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 $n\,\to \,\infty$ .