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Improvements of lower bounds for the least common multiple of finite arithmetic progressions. (English) Zbl 1157.11001

Let u 0 , r and n be positive integers with (u 0 ,r)=1. Let u k =u 0 +kr for 1kn and L n =lcm(u 0 ,u 1 ,,u n ). In this paper, the authors prove (by elementatry means) that L n u 0 r α (r+1) n if n>r α .

This result in case α=0 was conjectured by B. Farhi [C. R., Math., Acad. Sci. Paris 341, No. 8, 469–474 (2005; Zbl 1117.11005)] and proved by S. Hong and W. Feng [C. R., Math., Acad. Sci. Paris 343, No. 11–12, 695–698 (2006; Zbl 1156.11004)]. In the same paper they also proved the above result in case α=1.

11A05Multiplicative structure of the integers
11B25Arithmetic progressions