MathLabs

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 4 of 5: Check the remaining three roots
Previous stepNext step
x=−1⇒n=1−m,x=2,−2⇒(m,n)=(−4,−4),(−5,−6),(−6,−5),(3,−2),(−2,3)x=-1\Rightarrow n=1-m,\qquad x=2,-2\Rightarrow (m,n)=(-4,-4),(-5,-6),(-6,-5),(3,-2),(-2,3)
Detailed analysis

For x=-1, substitution gives n=1-m and again both expressions are squares, the same family in the opposite ordering. For x=2 and x=-2, substitution reduces the second square condition to elementary difference-of-squares equations; under |m|>=|n| the surviving pairs are (-4,-4), (-5,-6), and (3,-2). Symmetry adds (-6,-5) and (-2,3).