Fibonacci squares?

Number Theory Level pending

The Fibonacci numbers are given by:

$0,1,1,2,3,5,8,13,21....$

And in general they satisfy the following recursion relation:

For how many Fibonacci numbers does $F(n) = n^2$ ?

3 None of these 4 2 0 1

\infty

1 solution

Geoff Pilling
Nov 24, 2016

$F(n) = n^2$ for only $\boxed3$ values of $n$ , namely $0$ , $1$ , and $12$ .

There are none greater than $12$ , since the Fibonacci sequence increases much faster than $n^2$ for $n>12$ .