So there was one problem on the Purple Comet, which I mostly solved up to one point:
Find a three-digit positive integer $n$ and a positive integer $z$ that satisfy the equation $(n-3)(n-5)=6z^2$.
The answer is $n = 489$, but how do I get it from the equation?
I tried to solve like this:
Let’s say $n-3$ is $p+2$, and $n-5$ is $p$:
$$p(p+2)=6z^2$$
THEN
$$(p+1)^2=6z^2+1$$
I found that $z$ can be $2$, or $20$, when I tried.
(P.S. $n$ has to be a $3$ digit number that follows that rule.)
Hope to get some help! THANKS!