Hong, Shaofang; Kominers, Scott Duke Further improvements of lower bounds for the least common multiples of arithmetic progressions. (English) Zbl 1196.11007 Proc. Am. Math. Soc. 138, No. 3, 809-813 (2010). For relatively prime positive integers \(u_0\) and \(r\), the authors consider the arithmetic progression \( \{u_k:=u_0+kr\}_{k=0}^n\). Define \(L_n:=\text{lcm}\{u_0, u_1, \dots , u_n\}\) and let \( a\geq 2\) be any integer. In this paper they show that for integers \( \alpha, r\geq a\) and \( n\geq 2\alpha r\) \[ L_n\geq u_0r^{\alpha +a-2}(r+1)^n. \] In particular, letting \( a=2\) yields an improvement to the best previous lower bound on \( L_n\) (obtained by S. Hong and Y. Yang [Proc. Am. Math. Soc. 136, No. 12, 4111–4114 (2008; Zbl 1157.11001)]) for all but three choices of \( \alpha , r\geq 2\). Reviewer: Alexey Ustinov (Khabarovsk) Cited in 1 ReviewCited in 9 Documents MSC: 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11B25 Arithmetic progressions Keywords:lower bounds; least common multiple; arithmetic progression Citations:Zbl 1157.11001 PDFBibTeX XMLCite \textit{S. Hong} and \textit{S. D. Kominers}, Proc. Am. Math. Soc. 138, No. 3, 809--813 (2010; Zbl 1196.11007) Full Text: DOI arXiv References: [1] P. Bateman, J. Kalb, and A. Stenger, Problem 10797: A limit involving least common multiples, Amer. Math. Monthly 109 (2002), 393-394. · Zbl 1124.11300 [2] 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). · Zbl 1117.11005 · doi:10.1016/j.crma.2005.09.019 [3] 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. · Zbl 1124.11005 · doi:10.1016/j.jnt.2006.10.017 [4] Bakir Farhi and Daniel Kane, New results on the least common multiple of consecutive integers, Proc. Amer. Math. Soc. 137 (2009), no. 6, 1933 – 1939. · Zbl 1229.11007 [5] Denis Hanson, On the product of the primes, Canad. Math. Bull. 15 (1972), 33 – 37. · Zbl 0231.10008 · doi:10.4153/CMB-1972-007-7 [6] 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). · Zbl 1156.11004 · doi:10.1016/j.crma.2006.11.002 [7] S. Hong and G. Qian, The least common multiple of consecutive terms in arithmetic progressions, arXiv:0903.0530, 2009. [8] Shaofang Hong and Yujuan Yang, Improvements of lower bounds for the least common multiple of finite arithmetic progressions, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4111 – 4114. · Zbl 1157.11001 [9] Shaofang Hong and Yujuan Yang, On the periodicity of an arithmetical function, C. R. Math. Acad. Sci. Paris 346 (2008), no. 13-14, 717 – 721 (English, with English and French summaries). · Zbl 1213.11014 · doi:10.1016/j.crma.2008.05.019 [10] M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982), no. 2, 126 – 129. · Zbl 0494.10004 · doi:10.2307/2320934 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.