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Interesting problem

Source: EGMO 2018 Moldova TST

February 11, 2018

Problem Statement

Let SS = { x1x_1 , x2x_2 } be the solutions of the equation x2−2∗a∗x−1=0x^2-2*a*x -1 = 0 , where aa is a positive integer.Prove that for any n∈N n \in\mathbb{N} the expression E=18 E=\frac{1}{8}(x12n−x22nx_1^{2n}-x_2^{2n})(x14n−x24nx_1^{4n}-x_2^{4n}) is a product of consecutive numbers.
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