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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 :=lcm{u 0 ,u 1 ,,u n } and let a2 be any integer. In this paper they show that for integers α,ra and n2αr

L n u 0 r α+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 α,r2.

MSC:
11A05Multiplicative structure of the integers
11B25Arithmetic progressions