MathLabs

Problem 2

Find all positive integers a,ba,b such that a2+bb2−a\frac{a^2+b}{b^2-a} and b2+aa2−b\frac{b^2+a}{a^2-b} are both integers.
Step 6 of 7: Finish the adjacent case
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b=2,3,4,5⟹(a,b)=(3,2) only.b=2,3,4,5\Longrightarrow(a,b)=(3,2)\text{ only}.
Detailed analysis

Directly checking b=2,3,4,5 leaves only (a,b)=(3,2); the second fraction is 1 in every adjacent case, so both conditions are satisfied there.