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 3 of 5: Handle the root 1
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x=1⇒n=m+1,(m2+4n,n2+4m)=((m+2)2,(m+1)2)x=1\Rightarrow n=m+1,\qquad (m^2+4n,n^2+4m)=((m+2)^2,(m+1)^2)
Detailed analysis

Substituting x=1 into x squared plus m x minus n=0 gives n=m+1. Then both required expressions are automatically squares: (m+2) squared and (m+1) squared. This gives (m,n)=(k,1-k) after relabeling the free integer k.