The title says it all: Let $x$ and $y$ be integers, with $x$ even and $y$ odd, such that $x > y \ge 1$ and
$$x^4+4x^3y-4xy^3-4y^4 = w^2$$
for some integer $w$. Does this force $y \mid x$?
Brute force calculations seem to support that conjecture. In fact, they suggest the stronger conjecture $x=2y$.
I’ve poked around trying to prove something, but can’t figure out the magic incantation. Any advice/help would be appreciated.
Best Answer
It seems that the pair $x=951017531851446281396498, y=780923760568941116026369$ is a counterexample.