Lower bounds for the least common multiple of finite arithmetic progressions

Shaofang Hong; Weiduan Feng

doi:10.1016/j.crma.2006.11.002

Number Theory

Lower bounds for the least common multiple of finite arithmetic progressions

Presented by: Christophe Soulé

Shaofang Hong ¹; Weiduan Feng ¹

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

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

Abstract
Résumé

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 $L_{n} ⩾ u_{0} r {(r + 1)}^{n}$ .

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 $L_{n} ⩾ u_{0} r {(r + 1)}^{n}$ .

Received: 2006-09-14
Accepted: 2006-10-27
Published online: 2006-12-01

DOI: 10.1016/j.crma.2006.11.002

Author's affiliations:

Shaofang Hong ¹; Weiduan Feng ¹

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

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Shaofang Hong; Weiduan Feng. Lower bounds for the least common multiple of finite arithmetic progressions. Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 695-698. doi : 10.1016/j.crma.2006.11.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.002/

References
Cited by

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

[2] 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

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

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

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

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

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

Cited by Sources:

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

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