A number theory problem by Francisco Rodríguez

Given the Fibonacci sequence \(\{F_n\}=\{1, 1, 2, 3, 5, 8, \ldots\}\)

Find all the couples \(\{u,v\}\) of real positive numbers such that


Is an element of the Fibonacci sequence

Enter your answer as the sum \(u+v\) of all the couples \(\{u,v\}\) that satisfies the given condition.


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