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

Shaofang Hong; Yuanyuan Luo; Guoyou Qian; Chunlin Wang

doi:10.1016/j.crma.2013.10.005

Number theory

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

Presented by: Jean-Pierre Serre

Shaofang Hong ^1,²; Yuanyuan Luo ¹; Guoyou Qian ³; Chunlin Wang 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, Volume 351 (2013) no. 21-22, pp. 781-785.

Abstract
Résumé

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.

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.

Received: 2013-08-28
Accepted: 2013-10-09
Published online: 2013-10-26

DOI: 10.1016/j.crma.2013.10.005

Author's affiliations:

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

@article{CRMATH_2013__351_21-22_781_0,
     author = {Shaofang Hong and Yuanyuan Luo and Guoyou Qian and Chunlin Wang},
     title = {Uniform lower bound for the least common multiple of a polynomial sequence},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {781--785},
     publisher = {Elsevier},
     volume = {351},
     number = {21-22},
     year = {2013},
     doi = {10.1016/j.crma.2013.10.005},
     language = {en},
}

TY  - JOUR
AU  - Shaofang Hong
AU  - Yuanyuan Luo
AU  - Guoyou Qian
AU  - Chunlin Wang
TI  - Uniform lower bound for the least common multiple of a polynomial sequence
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 781
EP  - 785
VL  - 351
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2013.10.005
LA  - en
ID  - CRMATH_2013__351_21-22_781_0
ER  -

%0 Journal Article
%A Shaofang Hong
%A Yuanyuan Luo
%A Guoyou Qian
%A Chunlin Wang
%T Uniform lower bound for the least common multiple of a polynomial sequence
%J Comptes Rendus. Mathématique
%D 2013
%P 781-785
%V 351
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2013.10.005
%G en
%F CRMATH_2013__351_21-22_781_0

Shaofang Hong; Yuanyuan Luo; Guoyou Qian; Chunlin Wang. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.10.005/

References
Cited by

[1] S. Alaca; K.S. Williams Introductory Algebraic Number Theory, Cambridge University Press, Cambridge, 2004

[2] P. Bateman; J. Kalb; A. Stenger A limit involving least common multiples, Amer. Math. Monthly, Volume 109 (2002), pp. 393-394

[3] P.L. Chebyshev Memoire sur les nombres premiers, J. Math. Pures Appl., Volume 17 (1852), pp. 366-390

[4] B. Farhi Minoration non triviales du plus petit commun multiple de certaines suites finies dʼentiers, C. R. Acad. Sci. Paris, Ser. I, Volume 341 (2005), pp. 469-474

[5] B. Farhi Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory, Volume 125 (2007), pp. 393-411

[6] B. Farhi; D. Kane New results on the least common multiple of consecutive integers, Proc. Amer. Math. Soc., Volume 137 (2009), pp. 1933-1939

[7] D. Hanson On the product of the primes, Canad. Math. Bull., Volume 15 (1972), pp. 33-37

[8] S. Hong; W. Feng Lower bounds for the least common multiple of finite arithmetic progressions, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 695-698

[9] S. Hong; S.D. Kominers Further improvements of lower bounds for the least common multiple of arithmetic progressions, Proc. Amer. Math. Soc., Volume 138 (2010), pp. 809-813

[10] S. Hong; G. Qian The least common multiple of consecutive arithmetic progression terms, Proc. Edinb. Math. Soc., Volume 54 (2011), pp. 431-441

[11] S. Hong; G. Qian; Q. Tan The least common multiple of a sequence of products of linear polynomials, Acta Math. Hung., Volume 135 (2012), pp. 160-167

[12] M. Nair On Chebyshev-type inequalities for primes, Amer. Math. Monthly, Volume 89 (1982), pp. 126-129

[13] S.M. Oon Note on the lower bound of least common multiple, Abstr. Appl. Anal. (2013) (Art. ID 218125, 4 p)

[14] G. Qian; S. Hong Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms, Arch. Math., Volume 100 (2013), pp. 337-345

[15] R. Wu; Q. Tan; S. Hong New lower bounds for the least common multiples of arithmetic progressions, Chin. Ann. Math., Ser. B (2013) | DOI

Cited by 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.

Comments - Policy