Improvements of lower bounds for the least common multiple of finite arithmetic progressions

Hong, Shaofang; Yang, Yujuan

doi:10.1090/S0002-9939-08-09565-8

Improvements of lower bounds for the least common multiple of finite arithmetic progressions
HTML articles powered by AMS MathViewer

by Shaofang Hong and Yujuan Yang PDF

Proc. Amer. Math. Soc. 136 (2008), 4111-4114 Request permission

Abstract:

Let $u_0, r, \alpha$ and $n$ be positive integers such that $(u_ 0,r)=1$. Let $u_k=u_0+kr$ for $1\leq k\leq n$. We prove that $L_n :=\textrm {lcm}\{u_0, u_1,\cdots , u_n\}\geq u_ 0r^\alpha (r+1)^n$ if $n>r^\alpha$. This improves the lower bound of $L_n$ obtained previously by Farhi, Hong and Feng.

References

Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
Gennady Bachman and Troy Kessler, On divisibility properties of certain multinomial coefficients. II, J. Number Theory 106 (2004), no. 1, 1–12. MR 2029778, DOI 10.1016/j.jnt.2003.12.001
Bakir Farhi, Minorations non triviales du plus petit commun multiple de certaines suites finies d’entiers, C. R. Math. Acad. Sci. Paris 341 (2005), no. 8, 469–474 (French, with English and French summaries). MR 2180812, DOI 10.1016/j.crma.2005.09.019
Bakir Farhi, Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory 125 (2007), no. 2, 393–411. MR 2332595, DOI 10.1016/j.jnt.2006.10.017
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), 481-548.
Denis Hanson, On the product of the primes, Canad. Math. Bull. 15 (1972), 33–37. MR 313179, DOI 10.4153/CMB-1972-007-7
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
Shaofang Hong and Weiduan Feng, Lower bounds for the least common multiple of finite arithmetic progressions, C. R. Math. Acad. Sci. Paris 343 (2006), no. 11-12, 695–698 (English, with English and French summaries). MR 2284695, DOI 10.1016/j.crma.2006.11.002
Shaofang Hong and Raphael Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasg. Math. J. 46 (2004), no. 3, 551–569. MR 2094810, DOI 10.1017/S0017089504001995
Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
G. Myerson and J. W. Sander, What the least common multiple divides. II, J. Number Theory 61 (1996), no. 1, 67–84. MR 1418320, DOI 10.1006/jnth.1996.0138
M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982), no. 2, 126–129. MR 643279, DOI 10.2307/2320934