I need help in proving one elementary result with Fibonacci numbers. Prove that for $n>2$, the product $F_1 \cdot F_2 \cdots F_n$ cannot be a perfect square, where $F_1 = F_2 = 1, F_{n+1}=F_n + F_{n1}$.
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Take the largest prime $p$ up to $n$ (which is greater than $n/2$ by Bertrand's postulate) and notice that since the Fibonacci sequence is a divisibility sequence: $gcd(F_p,F_k)=F_{gcd(p,k)}=1$ for $k\neq p$ then if $\prod F_k$ is a perfect square so is $F_p$, but according to this theorem the only nontrivial square in the Fibonacci sequence is 144 and so you get the result. Remark: The paper "Diophantine equations with products of consecutive terms in Lucas sequences" by F. Luca, T.N. Shorey determines all products of consecutive Fibonacci numbers which are perfect powers. 

