Number theory
Uniform lower bound for the least common multiple of a polynomial sequence
Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 781-785.

Let n be a positive integer and f(x) be a polynomial with nonnegative integer coefficients. We prove that lcmn/2in{f(i)}2n, except that f(x)=x and n=1,2,3,4,6 and that f(x)=xs, with s2 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.

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 ppcmn/2in{f(i)}2n, 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.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2013.10.005
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
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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, Volume 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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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.