Given $p_1|(p_2-1)(p_2+1)$, is $p_1<p_2$? (for prime numbers, but possibly for all natural numbers)

Question

$p_1|(p_2-1)(p_2+1)$ means $$p_1\le\frac{p_2^2-1}{2}<p_2^2$$ because the greatest prime factor of a number can't be greater than half that number. But is $p_1<p_2$ true or not? I can't find any counter example, and intuitively that statement is true. But how would I prove it?

Answer 1

The answer is no. Consider $p_1=3$ and $p_2=2$. However, if $p_1$ and $p_2$ are odd, then it is true, as $p_1$ must divide either $\frac{p_2-1}2$ or $\frac{p_2+1}2$.