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Are there Fibonacci numbers other than $F_0 = 0 = 0^2, F_1 = F_2 = 1 = 1^2,$ and $F_{12} = 144 = 12^2$ which are square numbers? If not, what is the proof?

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Here is a paper of Bugeaud, Mignotte, and Siksek proving that

the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144

Therefore the only squares are 0, 1, and 144.

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0, 1 and 144 are the only perfect squares. See the proof here.

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  • $\begingroup$ Nice! Unlike the proof from the other answer, this one is elementary. Indeed, the proof you linked to is cited in Bugeaud, Mignotte and Siksek's paper. $\endgroup$ – John Gowers Jan 17 at 20:33

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