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Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
Published online by Cambridge University Press: 20 November 2018
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
$n\,\to \,\infty$.
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- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 2014
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