Suppose $1<p_1<\dots<p_n$ are pairwise relatively prime integers and let $\Delta=p_1\cdots p_n$. In the proof of Lemma 4.8 of this paper (https://arxiv.org/pdf/1606.08656.pdf), there is the following paragraph.

If $$D=\frac{3p_n\pm \sqrt{5p_n^2-4}}{2}\cdot \frac{\Delta}{p_n} $$ is an integer then $p_n$ is an odd-indexed Fibonacci number. In this case $\sqrt{5p_n^2-4} $ is an odd-indexed Lucas number and $D=p_1\cdots p_{n-1}F_{2m-1}$. Note that $D=\frac{3p_n- \sqrt{5p_n^2-4}}{2p_n}\Delta<\frac{2}{3}\Delta$.

Actually the paper is about symplectic embeddings of rational homology balls into $\Bbb CP^2$, and I can't understand most of this paragraph. I only know the definitions of Fibonacci and Lucas numbers.

Why is $p_n$ an odd-indexed Fibonacci number $F_{2m+1}$ (assuming $D$ is an integer)?

In this case why is $\sqrt{5p_n^2-4} $ an odd-indexed Lucas number?

Also how do we have $D=p_1\cdots p_{n-1}F_{2m-1}$?

In the last sentence how the sign $\pm$ in $D$ became $-$?