<spanclass=′latex−bold′>N6.</span> Let a,b be positive integers. If a,b satisfy that
\begin{align*} \frac{a+1}{b} + \frac{b+1}{a} \end{align*}
is also a positive integer, show that
\begin{align*} \frac{a+b}{gcd(a,b)^2} \end{align*}
is a Fibonacci number.
Proposed by usjl number theorygreatest common divisorIMOC