Perfect Squares in Fibonacci Numbers

Question

Let us define Fibonacci Numbers as: $F_1=F_2=1,F_n=F_{n-1}+F_{n-2}$ for $n>2$.

$F_1$,$F_2$, and $F_{12}$ are perfect squares.

Find the least integer $n>12$, if any, such that $F_{n}$ is a perfect square.

If there is no integer $n>12$ for which $F_n$ is a perfect square, then prove that.

I tried using Microsoft Excel, I found that $F_{51}$ is close to be a perfect square, but it is not.

I tried to use the parabola $y=x^2-x-1$ (the roots of which are the golden ratios $1.618...$ and $-0.618...$) somehow, but it was not helpful for me.

score 1 · Answer 1 · answered Jun 26 '19 at 22:27

The only Fibonacci squares are $0$, $1$, and $144$. This result was published in 1964 by J. H. E. Cohn.

If you would like to download the paper, you may find it here:

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