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

Details 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

2010-12-30

arxiv.org/abs/1101.0142v1

Mathematics

Number Theory

7 pages

Scientific paper

For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n := lcm(u_0, u_1, ..., u_n)$ of the finite arithmetic progression $u_k := u_0+kr$ ($k = 0, ..., n$). We derive new lower bounds on $L_n$ which 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 $n$ grows large.

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