# 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

$$u(F_n)^2+v(F_{n+1})^2$$

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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