Number Theory
Lower bounds for the least common multiple of finite arithmetic progressions
Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 695-698.

Let $u0,r$ and n be positive integers such that $(u0,r)=1$. Let $uk=u0+kr$ for $1⩽k⩽n$. We prove that $Ln:=lcm{u0,u1,…,un}⩾u0(r+1)n$ which confirms Farhi's conjecture (2005). Further we show that if $r, then $Ln⩾u0r(r+1)n$.

Soit $u0$, r et n des entiers positifs tels que $(u0,r)=1$, posons $uk=u0+kr$ pour $1⩽k⩽n$. Nous démontrons $Ln:=ppcm(u0,u1,…,un)⩾u0(r+1)n$, ce qui confirme la conjecture de Fahri (2005). De plus, nous montrons que si $r alors $Ln⩾u0r(r+1)n$.

Accepted:
Published online:
DOI: 10.1016/j.crma.2006.11.002
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/

[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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