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 5 of 5: State the complete answer
(m,n)=(k2,0),(0,k2),(k,1−k),(1−k,k),(−4,−4),(−5,−6),(−6,−5)(m,n)=(k^2,0),(0,k^2),(k,1-k),(1-k,k),(-4,-4),(-5,-6),(-6,-5)
Detailed analysis

Testing the listed families verifies both square conditions, and the small-root classification proves no other cases exist. Thus k is any integer in the two linear families and any integer in the square families.