Nair’s and Farhi’s Identities Involving the Least Common Multiple of Binomial Coefficients Are Equivalent

Throughout this note, let N denote the set of nonnegative integers. Define N∗ := N \ {0}. There are lots of known results about the least common multiple of a sequence of positive integers. The most renowned is nothing else than an equivalent of the prime number theory; it says that log lcm(1, 2, ..., n) ∼ n as n approaches infinity (see, for instance [6]), where lcm(1, 2, · · · , n) means the least common multiple of 1, 2, ..., n. Some authors found effective bounds for lcm(1, 2, ..., n). Hanson [5] got the upper bound lcm(1, 2, ..., n) ≤ 3(∀n ≥ 1). Nair [12] obtained the lower bound lcm(1, 2, · · · , n) ≥ 2(∀n ≥ 9). Nair [12] also gave a new nice proof for the well-known estimate lcm(1, 2, · · · , n) ≥ 2(∀n ≥ 1). Hong and Feng [7] extended this inequality to the general arithmetic progression, which confirmed Farhi’s conjecture [2]. Regarding to many other related questions and generalizations of the above results investigated by several authors, we refer the interested reader to [1], [4], [8]-[10]. By exploiting the integral ∫ 1 0 x(1− x)dx, Nair [12] showed the following identity involving the binomial coefficients: