On The Least Common Multiple Of Finite Arithmetic Progression

In 1972, Hanson [5] obtained an upper bound of lcm(1,...,n) whilein 1982, Nair [11] got a lower bound of lcm(1,...,n). In 2005, Farhi [2] obtained somenon-trivial lower bounds for the least common multiple of finite arithmetic progressions.Meanwhile Farhi conjectured that if u₀,r,n∈Z⁺, (u₀,r) = 1 and u_k = u₀ + kr for1≤k≤n, then L_n := lcm(u₀,u₁,...,u_n)≥u₀(r + 1)n. In order to investigate theinteger lcm(n,n +1,...,n+k), Farhi introduced the arithmetic function gk defined byg_k(n) := n(n + 1)···(n + k)/lcm(n,n + 1,...,n + k). Farhi pointed out that k! is aperiod of g_k. Furthermore, Farhi asked the question that for any given integer k≥0,what is the smallest positive period of the function gk? In this paper, we introduce anew method to show that Farhi's conjecture is true. Under the condition r < n, weget an improved lower bound L_n≥u₀r(r + 1)n. In the last part of this paper, we getseveral periods of gk which is strictly less than k!.