对任意正整数 $n$, 著名的F.Smarandache LCM 函数 $SL(n)$ 定义为最小的 正整数 $k$, 使得 $n|[1,2,\dots ,k]$, 这里 $[1,2,\dots ,k]$ 表示 $1,2,\dots ,k$ 的最小公倍数. 本文利用初等方法研究张文鹏在他所著的《初等数论》一书中提出的一个包含 F.Smarandache LCM 函数的猜想, 并有了实质性进展.

For any positive integer $n$, the famous F.Smarandache LCM function $SL(n)$ is defined as the smallest positive integer $k$ such that $n|[1,2,\dots ,k]$, where $[1,2,\dots ,k]$ denotes the least common multiple of $1,2,\dots ,k$. The main purpose of this paper is to use using the elementary methods to study a conjecture involving the F.Smarandache LCM function, which is proposed by Zhang Wenpeng in his book ``Elementary Number Theory'', thus making some substantial progress for this conjecture.