An identity involving the least common multiple of binomial coefficients and its application

Details An identity involving the least common multiple of binomial coefficients and its application An identity involving the least common multiple of binomial coefficients and its application

2009-06-12

arxiv.org/abs/0906.2295v2

Mathematics

Number Theory

5 pages. To appear in American Mathematical Monthly

Scientific paper

In this paper, we prove the identity $$\lcm\{\binom{k}{0}, \binom{k}{1},
>..., \binom{k}{k}\} = \frac{\lcm(1, 2, ..., k, k + 1)}{k + 1} (\forall k \in
\mathbb{N}) .$$
As an application, we give an easily proof of the well-known nontrivial lower
bound $\lcm(1, 2, ..., k) \geq 2^{k - 1}$ $(\forall k \geq 1)$.

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