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Zbl 1196.11007
Hong, Shaofang; Kominers, Scott Duke
Further improvements of lower bounds for the least common multiples of arithmetic progressions.
(English)
[J] Proc. Am. Math. Soc. 138, No. 3, 809-813 (2010). ISSN 0002-9939; ISSN 1088-6826/e

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 $a\ge 2$ 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 {\it S. Hong} and {\it Y. Yang} [Proc. Am. Math. Soc. 136, No. 12, 4111--4114 (2008; Zbl 1157.11001)]) for all but three choices of $\alpha , r\geq 2$.
[Alexey Ustinov (Khabarovsk)]
MSC 2000:
*11A05 Multiplicative structure of the integers
11B25 Arithmetic progressions

Keywords: least common multiple; arithmetic progression

Citations: Zbl 1157.11001

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