Lower bounds for the least common multiple of finite arithmetic progressions

Hong, Shaofang; Feng, Weiduan

doi:10.1016/j.crma.2006.11.002

Number Theory

Lower bounds for the least common multiple of finite arithmetic progressions
[Minoration du plus petit commun multiple d'une progression arithmétique finie]

Présenté par : Christophe Soulé

Hong, Shaofang ¹ ; Feng, Weiduan ¹

¹ Mathematical College, Sichuan University, Chengdu 610064, P.R. China

Comptes Rendus. Mathématique, Tome 343 (2006) no. 11-12, pp. 695-698.

Résumé
Abstract

Soit $u_{0}$ , r et n des entiers positifs tels que $(u_{0}, r) = 1$ , posons $u_{k} = u_{0} + k r$ pour $1 ⩽ k ⩽ n$ . Nous démontrons $L_{n} : = p p c m (u_{0}, u_{1}, \dots, u_{n}) ⩾ u_{0} {(r + 1)}^{n}$ , ce qui confirme la conjecture de Fahri (2005). De plus, nous montrons que si $r < n$ alors $L_{n} ⩾ u_{0} r {(r + 1)}^{n}$ .

Let $u_{0}, r$ and n be positive integers such that $(u_{0}, r) = 1$ . Let $u_{k} = u_{0} + k r$ for $1 ⩽ k ⩽ n$ . We prove that $L_{n} : = lcm {u_{0}, u_{1}, \dots, u_{n}} ⩾ u_{0} {(r + 1)}^{n}$ which confirms Farhi's conjecture (2005). Further we show that if $r < n$ , then $L_{n} ⩾ u_{0} r {(r + 1)}^{n}$ .

Reçu le : 2006-09-14
Accepté le : 2006-10-27
Publié le : 2006-12-01

DOI : 10.1016/j.crma.2006.11.002

Affiliations des auteurs :

Hong, Shaofang ¹ ; Feng, Weiduan ¹

¹ Mathematical College, Sichuan University, Chengdu 610064, P.R. China

@article{CRMATH_2006__343_11-12_695_0,
     author = {Hong, Shaofang and Feng, Weiduan},
     title = {Lower bounds for the least common multiple of finite arithmetic progressions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {695--698},
     publisher = {Elsevier},
     volume = {343},
     number = {11-12},
     year = {2006},
     doi = {10.1016/j.crma.2006.11.002},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.11.002/}
}

TY  - JOUR
AU  - Hong, Shaofang
AU  - Feng, Weiduan
TI  - Lower bounds for the least common multiple of finite arithmetic progressions
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 695
EP  - 698
VL  - 343
IS  - 11-12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2006.11.002/
DO  - 10.1016/j.crma.2006.11.002
LA  - en
ID  - CRMATH_2006__343_11-12_695_0
ER  -

%0 Journal Article
%A Hong, Shaofang
%A Feng, Weiduan
%T Lower bounds for the least common multiple of finite arithmetic progressions
%J Comptes Rendus. Mathématique
%D 2006
%P 695-698
%V 343
%N 11-12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2006.11.002/
%R 10.1016/j.crma.2006.11.002
%G en
%F CRMATH_2006__343_11-12_695_0

Hong, Shaofang; Feng, Weiduan. Lower bounds for the least common multiple of finite arithmetic progressions. Comptes Rendus. Mathématique, Tome 343 (2006) no. 11-12, pp. 695-698. doi : 10.1016/j.crma.2006.11.002. http://www.numdam.org/articles/10.1016/j.crma.2006.11.002/

Bibliographie
Cité par

[1] Apostol, T.M. Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976

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

[3] B. Green, T. Tao, The primes contain arbitrarily long arithmetic progression, Ann. Math., in press

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

[5] Hardy, G.H.; Wright, E.M. An Introduction to the Theory of Numbers, Oxford University Press, London, 1960

[6] Hong, S.; Loewy, R. Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasgow Math. J., Volume 46 (2004), pp. 551-569

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

Cité par Sources :

^⁎ Research is partially supported by SRF for ROCS, SEM.