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 7 of 7: Restore symmetry
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(a,b)∈{(2,2),(3,3),(2,1),(3,2)}∪{(1,2),(2,3)}.(a,b)\in\{(2,2),(3,3),(2,1),(3,2)\}\cup\{(1,2),(2,3)\}.
Detailed analysis

Adding the swapped pairs gives all solutions: (2,2), (3,3), (1,2), (2,3), (2,1), and (3,2).