MathLabs

Problem 6

Let a,ba,b be positive integers such that ab+1ab+1 divides a2+b2a^2+b^2. Show that (a2+b2)/(ab+1)(a^2+b^2)/(ab+1) is the square of an integer.
Step 1 of 4: Fix the integer quotient and minimize
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a2+b2=k(ab+1),k=a2+b2ab+1∈Z>0a^2+b^2=k(ab+1),\qquad k=\frac{a^2+b^2}{ab+1}\in\mathbb Z_{>0}
Detailed analysis

Write a2+b2=k(ab+1)a^2+b^2=k(ab+1) with k=a2+b2ab+1∈Z>0k=\frac{a^2+b^2}{ab+1}\in\mathbb Z_{>0}. For this fixed k, among all positive integer solutions choose one for which min⁡(a,b)\min(a,b) is minimal. By symmetry assume b≤ab\le a.