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2022 Irish Math Olympiad, 9

9. Let [i]k[/i] be a positive integer and let $x_0, x_1, x_2, \cdots$ be an infinite sequence defined by the relationship $$x_0 = 0$$ $$x_1 = 1$$ $$x_{n+1} = kx_n +x_{n-1}$$ For all [i]n[/i] $\ge$ 1 (a) For the special case [i]k[/i] = 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for [i]n[/i] $\ge$ 2 (b) For the general case of integers [i]k[/i] $\ge$ 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for [i]n[/i] $\ge$ 2

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