Note on the lower bound of least common multiple
type: Item
instance of: scholarly article
title: Note on the lower bound of least common multiple
zbMATH Open document ID: 1267.11004
DOI: 10.1155/2013/218125
author: Shea-Ming Oon
published in: Abstract and Applied Analysis
publication date: 2013-05-08T00:00:00Z
review text: Summary: Consider a sequence of positive integers in arithmetic progression \(u_k = u_0 + kr\) with \((u_0, r) = 1\). Denote the least common multiple of \(u_0, \dots, u_n\) by \(L_n\). We show that if \(n \geq r^2 + r\), then \(L_n \geq u_0 r^{r+1}(r + 1)\), and we obtain optimum result on \(n\) in some cases for such estimate. Besides, for quadratic sequences \(m^2 + c, (m + 1)^2 + c, \dots, n^2 + c\), we also show that the least common multiple is at least \(2^n\) when \(m \leq \lceil n/2 \rceil\), which sharpens a recent result of \textit{B. Farhi} [C. R., Math., Acad. Sci. Paris 341, No. 8, 469--474 (2005; Zbl 1117.11005)].
Mathematics Subject Classification ID: 11A05
zbMATH DE Number: 6161313
zbMATH Keywords: sequence of positive integers in arithmetic progression
zbMATH Keywords: lower bound of least common multiple
Wikidata QID: Q58915591
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/218125
OpenAlex ID: W2063840320
cites work: The primes contain arbitrarily long arithmetic progressions
cites work: On the Product of the Primes
cites work: On Chebyshev-Type Inequalities for Primes
cites work: On the Number of Primes Less Than or Equal x
cites work: Nontrivial lower bounds for the least common multiple of some sequences of integers.
cites work: New results on the least common multiple of consecutive integers
cites work: The least common multiple of consecutive arithmetic progression terms
cites work: Improvements of lower bounds for the least common multiple of finite arithmetic progressions
cites work: Further improvements of lower bounds for the least common multiples of arithmetic progressions
cites work: Nontrivial lower bounds for the least common multiple of some finite sequences of integers
cites work: THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION
cites work: Lower bounds for the least common multiple of finite arithmetic progressions
description: scientific article; zbMATH DE number 6161313