I have the relation $x^2 + 4xy + y^2 = 1$. What I need to do is prove that it has infinitely many integer solutions.
I started out by solving for $y$ and getting $y = -2x \pm \sqrt{3x^2 + 1}$ and this shows that there are infinently many solutions as $x$'s domain is all reals, but I don't know how to prove that it has infinently many integer solutions. Any help would be awesome.
Thanks!
Best Answer
Completing the square, we get $$x^2+4xy+y^2=x^2+4xy^2+4y^2-3y^2=(x+2y)^2-3y^2=1.$$ Letting $z=x+2y$ (so $x=z-2y$), you now just have to show that there are infinitely many integer solutions to $$z^2-3y^2=1.$$ This is an instance of the famous Pell's equation (Wikipedia).