Two short remarks:

**1)** In the paper

> C. Ballot and F. Luca, ‘ On the
> equation $x^2 + dx^2 = F_n$ ’, Acta Arith.
> 127 (2007), 145–155.

the authors show that the equation $x^2+y^2=F_{2n}$ has no solution for most integers $n$.

**2)** In the problem section of the 42th volume of the [Fibonacci Quarterly][1], 2004, it is shown that the set of integers $n$ such that $F_n$ is divisible by a prime of the form $4k+3$ has asymptotic density $\frac12$.  


  [1]: http://www.fq.math.ca/42-1.html