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Lower bounds for the least common multiple of finite arithmetic progressions. (English) Zbl 1156.11004
Summary: Let $u_0$, $r$ and $n$ be positive integers such that $(u_0,r)=1.$ Let $u_k=u_0+kr$ for $1\leq k\leq n$. We prove that $L_n:=\text{lcm}\{u_0,u_1,\dots, u_n\}\geq u_0(r+1)^n$ which confirms {\it B. Farhi}’s conjecture [C. R., Math., Acad. Sci. Paris 341, No. 8, 469--474 (2005; Zbl 1117.11005)]. Further we show that if $r<n$, then $L_n\geq u_0r(r+1)^n$.

11A05Multiplicative structure of the integers
11B25Arithmetic progressions
Full Text: DOI
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