Review question

# Can we find a sequence of solutions to the equation $x^2 - 2y^2 = 1?$ Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8033

## Suggestion

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)?