A Slightly Less Pointless Solution to Le Monde puzzle #824

Here's Le Monde puzzle #824:

Show that, for any integer $y$, \[(\sqrt{3}-1)^{2y}+(\sqrt{3}+1)^{2y}\] is an integer multiple of a power of two.

Solution:

Consider \[f(n) = (-1+\sqrt{3})^{n}+(-1-\sqrt{3})^{n}\] and observe that the two bases are the roots of the quadratic $x^2 + 2x - 2$, hence $ f(n) $ obeys the recursion $ x_{n+2} = -2 x_{n+1} + 2 x_n $ with $ x_0=2$ and $x_1=-2 $. It follows that $ f(n) $ is always an even integer.

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