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Further improvements of lower bounds for the least common multiples of arithmetic progressions. (English) Zbl 1196.11007

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 \( \alpha , r\geq 2\).

MSC:

11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11B25 Arithmetic progressions

Citations:

Zbl 1157.11001
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References:

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