lean_workbook_plus_53179

First, we know $(m,n)=(3,2)$ is a solution. We can construct infinite solution as follows: Assume that $(m,n)$ is a solution.(So $gcd(m,2n)=1$ ) We can take positive integer $r$ such that $m^4-2n^4=r^2$ . Set $(x,y)$ $=$ ( $m^4+2n^4, 2mnr$ ). Then, you can easily confirm $gcd(x,2y)=1$ and $x^4-2y^4$ is a perfect square. You are done. (example: (x,y)=(113, 84), (262621633, 151245528))