Uniform lower bound for the least common multiple of a polynomial sequence

Hong, Shaofang; Luo, Yuanyuan; Qian, Guoyou; Wang, Chunlin

doi:10.1016/j.crma.2013.10.005

Number theory

Uniform lower bound for the least common multiple of a polynomial sequence
[Une borne inférieure uniforme pour le plus petit commun multiple dʼune suite polynomiale]

Présenté par : Jean-Pierre Serre

Hong, Shaofang ^1,² ; Luo, Yuanyuan ¹ ; Qian, Guoyou ³ ; Wang, Chunlin ¹

¹ Mathematical College, Sichuan University, Chengdu 610064, PR China
² Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, PR China
³ Center for Combinatorics, Nankai University, Tianjin 300071, PR China

Comptes Rendus. Mathématique, Tome 351 (2013) no. 21-22, pp. 781-785.

Résumé
Abstract

Soit n un entier ⩾1 et $f (x)$ un polynôme à coefficients entiers ⩾0. Nous démontrons que, à lʼexception de certains cas explicites, on a ${ppcm}_{⌈ n / 2 ⌉ ⩽ i ⩽ n} {f (i)} ⩾ 2^{n}$ , où $⌈ n / 2 ⌉$ dénote le plus petit entier $⩾ n / 2$ . Ceci améliore, et étend, les bornes inférieures obtenues par M. Nair en 1982, B. Farhi en 2007 et S.M. Oon en 2013.

Let n be a positive integer and $f (x)$ be a polynomial with nonnegative integer coefficients. We prove that ${lcm}_{⌈ n / 2 ⌉ ⩽ i ⩽ n} {f (i)} ⩾ 2^{n}$ , except that $f (x) = x$ and $n = 1, 2, 3, 4, 6$ and that $f (x) = x^{s}$ , with $s ⩾ 2$ being an integer and $n = 1$ , where $⌈ n / 2 ⌉$ denotes the smallest integer, which is not less than $n / 2$ . This improves and extends the lower bounds obtained by M. Nair in 1982, B. Farhi in 2007 and S.M. Oon in 2013.

Reçu le : 2013-08-28
Accepté le : 2013-10-09
Publié le : 2013-10-26

DOI : 10.1016/j.crma.2013.10.005

Affiliations des auteurs :

Hong, Shaofang ^{1, 2} ; Luo, Yuanyuan ¹ ; Qian, Guoyou ³ ; Wang, Chunlin ¹

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     title = {Uniform lower bound for the least common multiple of a polynomial sequence},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {781--785},
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     doi = {10.1016/j.crma.2013.10.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.10.005/}
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Hong, Shaofang; Luo, Yuanyuan; Qian, Guoyou; Wang, Chunlin. Uniform lower bound for the least common multiple of a polynomial sequence. Comptes Rendus. Mathématique, Tome 351 (2013) no. 21-22, pp. 781-785. doi : 10.1016/j.crma.2013.10.005. http://www.numdam.org/articles/10.1016/j.crma.2013.10.005/

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Cité par Sources :

^☆ The work was supported partially by National Science Foundation of China Grant #11371260, by the Ph.D. Programs Foundation of Ministry of Education of China Grant #20100181110073 and by Postdoctoral Science Foundation of China Grant #2013M530109.