Define the Fibonacci sequence
Then
as can be verified by induction (or otherwise).
Using this formula, we can check that
Hence the following relation is satisfied:
This is a quadratic equation in
; its discriminant is
when
k is even. This must be a perfect square since
is an integer, and since there are infinitely many Fibonacci numbers such that
k is even, there are infinitely many integers
n such that
is a perfect square.
Similarly the discriminant is
when
k is odd; this must be a perfect square, and as there are infinitely many Fibonacci numbers such that
k is odd, there are also infinitely many integers
n such that
is a perfect square.