Find the period of $f(x)$ with $f(x)f(y)=f(x+y)+f(x-y)$. For any real $x$, $\;\;y$
$f(x)f(y)=f(x+y)+f(x-y)$ with $f(1)=1$
Find the period of $f(x)$.




If $f(x)=2\cos\left(\frac{5\pi}{3}x\right)$, then $f$ satisfies the constraints of the problem. But $f(x+6/5)=f(x)$, so $6$ is not the fundamental period of $f$.
In general we could have $f(x)=2\cos\left(\frac{6n \pm 1}{3}\pi x\right)$ for any $n$, so the fundamental period of $f$ can be made arbitrarily small.
