On the periodicity of an arithmetical function

Shaofang Hong; Yujuan Yang

doi:10.1016/j.crma.2008.05.019

Number Theory

On the periodicity of an arithmetical function

Presented by: Christophe Soulé

Shaofang Hong ¹; Yujuan Yang ¹

¹ Mathematical College, Sichuan University, Chengdu 610064, PR China

Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 717-721.

Abstract
Résumé

Let $k ⩾ 0$ be an integer. When studying the least common multiple of $k + 1$ consecutive integers, Farhi introduced the arithmetical function $g_{k}$ defined for any positive integer n by $g_{k} (n) : = \frac{n (n + 1) \dots (n + k)}{lcm (n, n + 1, \dots, n + k)}$ . Farhi proved that $g_{k}$ is periodic and k! is a period of $g_{k}$ . Meanwhile Farhi raised an open problem determining the smallest positive period of $g_{k}$ . In this Note, we first show that $g_{k} (1) | g_{k} (n)$ for all positive integers n. Consequently, using this result, we show that for all positive integers k, $lcm (1, 2, \dots, k)$ is a period of $g_{k}$ , thus improving Farhi's result.

Soit $k ⩾ 0$ un entier, en étudiant le plus petit commun multiple de $k + 1$ entiers consécutifs Farhi a introduit la fonction arithmétique définie par $g_{k} (n) : = \frac{n (n + 1) \dots (n + k)}{ppcm (n, n + 1, \dots, n + k)}$ pour n entier positif. Farhi a démontré que $g_{k}$ est périodique et que k! en est une période. Dans le même temps Farhi a posé la question de déterminer la plus petite période de $g_{k}$ . Dans cette Note, nous démontrons pour commencer $g_{k} (1) | g_{k} (n)$ pour tout entier positif n. Puis, utilisant ce résultat, nous montrons que $ppcm (1, 2, \dots, k)$ est une période de $g_{k}$ pour tout entier positif k, ce qui améliore le résultat de Farhi.

Received: 2007-10-31
Accepted: 2008-05-26
Published online: 2008-07-01

DOI: 10.1016/j.crma.2008.05.019

Author's affiliations:

Shaofang Hong ¹; Yujuan Yang ¹

¹ Mathematical College, Sichuan University, Chengdu 610064, PR China

@article{CRMATH_2008__346_13-14_717_0,
     author = {Shaofang Hong and Yujuan Yang},
     title = {On the periodicity of an arithmetical function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {717--721},
     publisher = {Elsevier},
     volume = {346},
     number = {13-14},
     year = {2008},
     doi = {10.1016/j.crma.2008.05.019},
     language = {en},
}

TY  - JOUR
AU  - Shaofang Hong
AU  - Yujuan Yang
TI  - On the periodicity of an arithmetical function
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 717
EP  - 721
VL  - 346
IS  - 13-14
PB  - Elsevier
DO  - 10.1016/j.crma.2008.05.019
LA  - en
ID  - CRMATH_2008__346_13-14_717_0
ER  -

%0 Journal Article
%A Shaofang Hong
%A Yujuan Yang
%T On the periodicity of an arithmetical function
%J Comptes Rendus. Mathématique
%D 2008
%P 717-721
%V 346
%N 13-14
%I Elsevier
%R 10.1016/j.crma.2008.05.019
%G en
%F CRMATH_2008__346_13-14_717_0

Shaofang Hong; Yujuan Yang. On the periodicity of an arithmetical function. Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 717-721. doi : 10.1016/j.crma.2008.05.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.05.019/

References
Cited by

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

[2] G. Bachman; T. Kessler On divisibility properties of certain multinomial coefficients II, J. Number Theory, Volume 106 (2004), pp. 1-12

[3] B. Farhi 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

[4] 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

[5] B. Green; T. Tao The primes contain arbitrarily long arithmetic progression, Ann. of Math. (2), Volume 167 (2008), pp. 481-548

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

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

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

[10] S. Hong, Y. Yang, Improvements of lower bounds for the least common multiple of finite arithmetic progressions, Proc. Amer. Math. Soc., in press

[11] G. Myerson; J. Sander What the least common multiple divides II, J. Number Theory, Volume 61 (1996), pp. 67-84

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

Cited by Sources:

^⁎ This work was supported partially by Program for New Century Excellent Talents in University Grant # NCET-06-0785.

Comments - Policy