Problem 4
Find all pairs of integers (m,n) such that m squared plus 4n and n squared plus 4m are both perfect squares.
Step 1 of 5: Order the variables and form a quadratic
Detailed analysis
The problem is symmetric in m and n, so assume |m|>=|n|. If n=0, the first expression shows m is a square, giving (m,n)=(k squared,0); symmetry gives the other zero family. For n nonzero, the first expression being a square means x squared plus m x minus n has two nonzero integer roots x1,x2, with x1+x2=-m and x1x2=-n.