We’re all familiar with the Fibonacci sequence:
$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 \ldots$
I was messing around a little on a piece of paper and came to the conclusion that there wouldn’t be any numbers from the sequence that when raised to the power of two, becomes another number contained by the sequence—except that of $1^2$ however, but we’ll leave that out of it.
Although it occurred to me that there probably isn’t any number from the sequence that has this ability other than that of $1$, I don't know how to prove or disprove this.
Whilst I probably could write an application to retrieve the answer for me, I’m more interested in a mathematical solution. Should you clever people prove not to have a great solution, perhaps someone can point me in the right direction?