This is a curiosity question
I'm trying to solve a Diophantine equation and I need some results about fibonnacci numbers, I encountered this problem:
For which couple of integers $(n,m)$ the number $F_n^2+F_m^2$ is a square where $F_k$ denotes the $k$th Fibonacci number?
for example when $n$ and $m$ have opposite parity we have : $$ F_n^2+F_m^2=F_{n-m}F_{n+m} $$ and if we suppose $\gcd(F_n,F_m)=1$ we can conclude that $F_{n-m},F_{n+m}$ are squares, and we know that the only squares among Fibonacci numbers are $F_0=0,F_1=F_2=1$ and $F_{12}=144$
Fortunately there is a lot of proprieties related to Fibonacci numbers, but at the same time it's very difficult to find the appropriate method.
Question : Is there an efficient method to solve this problem?