I find the following statement about the homotopy equivalence of Lens spaces in (Wikipedia](https://en.wikipedia.org/wiki/Lens_space). The three-dimensional spaces $L(p,q_1)$ and $L(p,q_2)$ are homotopy equivalent if and only if $q_1 q_2\equiv \pm n^2 \ \textrm{mod} \ p$ for some integer $n$. I wonder if somebody tells me the concise proof or the reference for the proof.

I find the following statement about the homotopy equivalence of Lens spaces in Wikipedia. The three-dimensional spaces $L(p,q_1)$ and $L(p,q_2)$ are homotopy equivalent if and only if $q_1 q_2\equiv \pm n^2 \ \textrm{mod} \ p$ for some integer $n$. I wonder if somebody tells me the concise proof or the reference for the proof.