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 1 of 7: Use symmetry
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(a,b)↦(b,a) preserves the conditions;a≥b.(a,b)\mapsto(b,a)\text{ preserves the conditions};\quad a\ge b.
Detailed analysis

The two displayed divisibility conditions interchange when a and b are interchanged, so assume a is at least b.