Find $n \in \mathbb N$ such that $F_{n-1} \cdot x^2 - F_n \cdot y ^ 2 = (- 1) ^ n$ has a solution in positive integers $x, y$? 
Let $\{F_n\} -$ Fibonacci sequence: $F_1=F_2=1, F_{n+1}=F_n+F_{n-1}, n\ge2$. Find $n \in \mathbb N$  such that 
  $$F_{n-1} \cdot x^2 - F_n \cdot y ^ 2 = (- 1) ^ n$$ has a solution in positive integers $x, y$?

How to deal with do not know, for example, when $n = 3$, the solution is: $x = 7, y = 5.$
Addition
If $n=10:$
$$34\cdot538^2 - 55\cdot423^2 = 1.$$
 A: For a fixed number like $n\geq 3$, consider the following equation
$$
F_{n-1} \cdot u - F_n \cdot v = {(- 1)}^n\, \tag{1}
$$
where $F_n$ is the $nth$ Fibonacci numbers. Look at the equation $(1)$ as a Diophantus equation. We know that
$$
gcd(F_{n-1},F_n)=F_{gcd(n-1,n)}=F_1=1
$$
which results that 
$$
gcd(F_{n-1},F_n) \mid {(- 1)}^n
$$
It means the equation $(1)$ is solvable. One private solution of  equation $(1)$ is in the following form
$$
v_0=F_{n-2}  \quad ,  \quad u_0=F_{n-1}
$$
because by replacing $u_0$ and $v_0$ in  $(1)$, we have 
$$
F_{n-1}^2 - F_n \cdot F_{n-2} = {(- 1)}^n\, \tag{2}
$$
the relation $(2)$ is the Cassini formula in the case $n-1$. So the public solution of the equation $(1)$ is as follows 
$$
\left\{
\begin{array}{lcl}
v_t=v_0+\frac{F_{n-1}}{gcd(F_{n-1},F_n)}\,t &\Longrightarrow& v_t=F_{n-2}+F_{n-1}\,t \\
&& \tag{3}\\ 
u_t=u_0+\frac{F_n}{gcd(F_{n-1},F_n)}\,t &\Longrightarrow& u_t=F_{n-1}+F_n\,t 
\end{array}
\right.
$$
where $t$ is a natural number. 
Now for your question, we fix the number  $n$, and construct the
equation $(3)$ and find a value for the number $t$ such that $v_t$ and $u_t$ be square numbers. 
For example, suppose that $n=3$, then we have
$$
\left\{
\begin{array}{l}
v_t=1+t \\
u_t=1+2\,t
\end{array}
\right.
$$
the first value of $t$ that $v_t$ and $u_t$ are square numbers, is $t=24$. Or when we select $n=10$, the 
equation $(3)$ is as follows 
$$
\left\{
\begin{array}{l}
v_t=21+34\,t \\
u_t=34+55\,t
\end{array}
\right.
$$
we can see that the first $t$ that $v_t$ and $u_t$ are square numbers, is $t=5262$.
