Fibonacci Pairs Find all positive integer solutions to 
$y^2 - xy - x^2 = 1$ and  $y^2 - xy - x^2 = -1$
I have written a C++ program to yield some solution for large constants. I must make conjectures based on the output.
I notice that the output resembles the Fibonacci sequence.
My code http://cpp.sh/7qsjw
I know that the ratio between the $F_{n}$ and $F_{n-1}$ terms is approximately equal to $\frac{1+5^{1/2}}{2}$, the Golden Ratio.  Are there more conjectures to this problem.
 A: We use the indexing $F_0=0$, $F_1=1$ for the Fibonacci numbers. We want to show that 
$$F_{n+1}^2-F_{n+1}F_n-F_n^2=(-1)^n.$$
The result holds at $n=0$. Suppose it holds at $n=k$. We show the result holds at $n=k+1$. So the induction hypothesis is 
$$F_{k+1}^2-F_{k+1}F_k-F_k^2=(-1)^k,\tag{1}$$
and we want to show that
$$F_{k+2}^2-F_{k+2}F_{k+1}-F_{k+1}^2=(-1)^{k+1}\tag{2}.$$
In the left side of (2), replace $F_{k+2}$ by $F_{k+1}+F_k$. We obtain
$$(F_{k+1}+F_k)^2 -(F_{k+1}+F_k)F_{k+1}-F_{k+1}^2.\tag{3}$$
Expand and simplify. We get
$$F_k^2+F_{k+1}F_k-F_{k+1}^2,\quad\text{that is,}\quad
-(F_{k+1}^2-F_{k+1}F_k-F_k^2).$$
By the induction hypothesis this is $-(-1)^k$, which is $(-1)^{k+1}$. That completes the induction step.
Remark: We have shown that consecutive Fibonacci numbers alternately satisfy one of the two given equations. We have not shown that consecutive Fibonacci numbers are the only solutions of these equations in non-negative integers. But that is true, and follows from general facts about Pell-type equations.
