Note on the lower bound of least common multiple. (English) Zbl 1267.11004

Oon, Shea-Ming

Abstr. Appl. Anal. 2013, Article ID 218125, 4 p. (2013).

Summary: Consider a sequence of positive integers in arithmetic progression \(u_k = u_0 + kr\) with \((u_0, r) = 1\). Denote the least common multiple of \(u_0, \dots, u_n\) by \(L_n\). We show that if \(n \geq r^2 + r\), then \(L_n \geq u_0 r^{r+1}(r + 1)\), and we obtain optimum result on \(n\) in some cases for such estimate. Besides, for quadratic sequences \(m^2 + c, (m + 1)^2 + c, \dots, n^2 + c\), we also show that the least common multiple is at least \(2^n\) when \(m \leq \lceil n/2 \rceil\), which sharpens a recent result of B. Farhi [C. R., Math., Acad. Sci. Paris 341, No. 8, 469–474 (2005; Zbl 1117.11005)].

MSC:

11A05

Multiplicative structure; Euclidean algorithm; greatest common divisors

Keywords:

sequence of positive integers in arithmetic progression; lower bound of least common multiple

Citations:

Zbl 1117.11005

References:

[1]	Green, B.; Tao, T., The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics, 167, 2, 481-547 (2008) · Zbl 1191.11025 · doi:10.4007/annals.2008.167.481
[2]	Hanson, D., On the product of the primes, Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques, 15, 33-37 (1972) · Zbl 0231.10008
[3]	Nair, M., On Chebyshev-type inequalities for primes, The American Mathematical Monthly, 89, 2, 126-129 (1982) · Zbl 0494.10004 · doi:10.2307/2320934
[4]	Shapiro, H. N., On the number of primes less than or equal \(x\), Proceedings of the American Mathematical Society, 1, 346-348 (1950) · Zbl 0037.31403
[5]	Farhi, B., Minorations non triviales du plus petit commun multiple de certaines suites finies d’entiers, Comptes Rendus Mathématique. Académie des Sciences. Paris, 341, 8, 469-474 (2005) · Zbl 1117.11005 · doi:10.1016/j.crma.2005.09.019
[6]	Farhi, B.; Kane, D., New results on the least common multiple of consecutive integers, Proceedings of the American Mathematical Society, 137, 6, 1933-1939 (2009) · Zbl 1229.11007 · doi:10.1090/S0002-9939-08-09730-X
[7]	Hong, S.; Qian, G., The least common multiple of consecutive arithmetic progression terms, Proceedings of the Edinburgh Mathematical Society, 54, 2, 431-441 (2011) · Zbl 1304.11008 · doi:10.1017/S0013091509000431
[8]	Hong, S.; Yang, Y., Improvements of lower bounds for the least common multiple of finite arithmetic progressions, Proceedings of the American Mathematical Society, 136, 12, 4111-4114 (2008) · Zbl 1157.11001 · doi:10.1090/S0002-9939-08-09565-8
[9]	Hong, S.; Kominers, S. D., Further improvements of lower bounds for the least common multiples of arithmetic progressions, Proceedings of the American Mathematical Society, 138, 3, 809-813 (2010) · Zbl 1196.11007 · doi:10.1090/S0002-9939-09-10083-7
[10]	Wu, R.; Tan, Q.; Hong, S., New lower bounds for the least common multiple of arithmetic progressions
[11]	Farhi, B., Nontrivial lower bounds for the least common multiple of some finite sequences of integers, Journal of Number Theory, 125, 2, 393-411 (2007) · Zbl 1124.11005 · doi:10.1016/j.jnt.2006.10.017
[12]	Qian, G.; Tan, Q.; Hong, S., The least common multiple of consecutive terms in a quadratic progression, Bulletin of the Australian Mathematical Society, 86, 389-404 (2012) · Zbl 1290.11015
[13]	Hong, S.; Feng, W., Lower bounds for the least common multiple of finite arithmetic progressions, Comptes Rendus Mathématique. Académie des Sciences. Paris, 343, 11-12, 695-698 (2006) · Zbl 1156.11004 · doi:10.1016/j.crma.2006.11.002