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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asked Nov 9, 2014 at 8:02
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2 Answers
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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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$\begingroup$ 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. $\endgroup$ Commented Jan 17, 2019 at 20:33
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$\begingroup$ I think point (9) is false as written. $L_{-n} = (-1)^n L_n$ is true, but what's written is $L_{-n} = (-1)^{n-1}L_n$ notice the exponent. I'm still not sure if point (9) is used as written or if it is just incorrectly written but the true statement was used. $\endgroup$ Commented Jan 16, 2020 at 18:59
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$\begingroup$ The exponent in point (9) is just a transcription error. The original article in the Fibonacci Quarterly has the correct exponent. $\endgroup$– PhiraCommented Oct 18, 2022 at 9:09