<s>[INST] I want to find a pair of integers $X$, $Y$ which satisfy $X^2 - 2Y^2=1$ such that $X > Y > 50$ 
Find a pair of integers $X, Y$, which satisfy $X^2 - 2Y^2 = 1$, such that $X > Y > 50$. 

I have started by finding a pair of much smaller integers that work: $X(1) = 3$ and $Y(1) = 2$. 
When I looked up a solution it was as follows: 
$$\begin{array}
&X(2) = 3 × 3 + 4 × 2 = 17, & Y(2) = 2 × 3 + 3 × 2 = 12 \\
X(3) = 3 × 17 + 4 × 12 = 99, & Y(3) = 2 × 17 + 3 × 12 = 70
\end{array}$$
So $X = 99$ and $Y = 70$ is such a pair.
But what is the method used? I tried searching for similar questions, but didn't find an answer to this.
 [/INST] $$X_{i+1}^2-2Y_{i+1}^2 = (3X_i + 4Y_i)^2 - 2(2X_i + 3Y_i)^2 =$$ $$ = (9-2\cdot4)X_i ^ 2 + (12-12)X_i \cdot Y_i + (16-2\cdot9)Y_i ^ 2= $$ $$= X_{i}^2-2Y_{i}^2 = 1$$
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