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.
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9$\begingroup$ You might want to check Exercise 29 of Section 4.2 of Hatcher's algebraic topology. It gives a more general answer and a step-to-step guide of the proof. $\endgroup$– Mingcong ZengCommented Nov 8, 2015 at 23:28
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