1. Given integers \(a\), \(b\), we define two sequences \(x_1\), \(x_2\), \(x_3\), \(\dots\) and \(y_1\), \(y_2\), \(y_3\), \(\dots\) by setting \[x_{n+1} = 3x_n + 4y_n, \quad y_{n+1} = ax_n + by_n, \quad \text{for $n \ge 1$}.\]

Find positive values for \(a\), \(b\) such that \[(x_{n+1})^2 - 2(y_{n+1})^2 = (x_n)^2 - 2(y_n)^2.\]

Can you combine what we know to form an equation involving \(a\) and \(b\)?

  1. Find a pair of integers \(X\), \(Y\) which satisfy \(X^2 - 2Y^2 = 1\) such that \(X > Y > 50\).

How can we use what we have learnt from (i) and (ii)?