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perfect squares in a difference equation

Source: IrMO 2022

May 11, 2022
difference equationsalgebra

Problem Statement

9. Let k be a positive integer and let x0,x1,x2,⋯x_0, x_1, x_2, \cdots be an infinite sequence defined by the relationship x0=0x_0 = 0 x1=1x_1 = 1 xn+1=kxn+xn−1x_{n+1} = kx_n +x_{n-1} For all n ≥\ge 1 (a) For the special case k = 1, prove that xn−1xn+1x_{n-1}x_{n+1} is never a perfect square for n ≥\ge 2 (b) For the general case of integers k ≥\ge 1, prove that xn−1xn+1x_{n-1}x_{n+1} is never a perfect square for n ≥\ge 2
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