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

Abstract

For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\dots ,u_n)$ 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 $n\to \infty$.

arXiv comment10 pages

References in corpus (1)

• The least common multiple of consecutive arithmetic progression terms

Cited by in corpus (3)

• Nontrivial effective lower bounds for the least common multiple of some quadratic sequences
• Asymptotic behavior of the least common multiple of consecutive reducible quadratic progression terms
• Nontrivial effective lower bounds for the least common multiple of a $q$-arithmetic progression