Firstly, merry christmas!
I've got stuck at a problem.
If x, y are nonzero natural numbers with $x>y$ such that $$x-y = 5y^2 - 4x^2,$$ prove that $x - y$ is perfect square.
What I've thought so far: $$x - y = 4y^2 - 4x^2 + y^2$$ $$x - y = 4(y-x)(y+x) + y^2$$ $$x - y + 4(x-y)(x+y) = y^2$$ $$(x-y)(4x+4y+1) = y^2$$ So $4x+4y+1$ is a divisor of $y^2$.
I also take into consideration that $y^2$ modulo $4$ is $0$ or $1$ (I don't know if this can help.)
So how do I prove that $4x+4y+1$ is a perfect square (this would involve $x-y$ - a perfect square)? While taking examples, I couldn't find any perfect square with a divisor that is $M_4 + 1$ and is not perfect square.
If there are any mistakes or another way, please tell me.
Some help would be apreciated. Thanks!