## Wednesday, July 10, 2013

### 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.