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 4 of 4: Conclude that the quotient is a square
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C=0⟹AC=B2−k⟹k=B2C=0\quad\Longrightarrow\quad AC=B^2-k\quad\Longrightarrow\quad k=B^2
Detailed analysis

Substituting C=0C=0 into AC=B2−kAC=B^2-k gives k=B2k=B^2. Thus the required quotient k=a2+b2ab+1k=\frac{a^2+b^2}{ab+1} is the square of the integer B.