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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?

daOnlyBG
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user1321345589
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2 Answers2

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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.

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

bifurcat
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  • 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. – John Gowers Jan 17 '19 at 20:33
  • 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. – Enrico Borba Jan 16 '20 at 18:59
  • The exponent in point (9) is just a transcription error. The original article in the Fibonacci Quarterly has the correct exponent. – Phira Oct 18 '22 at 09:09