I love hate Fibonacci

The problem of Fibonacci squares is one of my favorites relating to Lucas sequences. There are only $3$ Fibonacci numbers that are also perfect squares, namely $0,$ $1,$ and $12.$ A proof from 1964 using Lucas numbers can be found here. So knowing that $0 = 0^2,$ $1 = 1^2,$ and $144 = 12^2,$ the only possible cases where $F_n = n^2$ are $n =0,$ $n = 1,$ and $n = 12.$ Indeed, $F_{12} = 144 = 12^2.$

We should think of Fibonacci as a friend. Look at all the mental exercise he gave us.