TITLE: About finding the period of a function which satisfies $f(x-2)+f(x+2)=f(x)$.
QUESTION [6 upvotes]: I had a problem with solving the following;
$f(x-2)+f(x+2)=f(x)$ I have tried to make it look like $f(x+T)=f(x)$ but I failed. I mostly have difficulty in solving these type of problems. What do you suggest I should do?

REPLY [5 votes]: Using this, the characteristic equation of $$f(x+2)-f(x)+f(x-2)=0$$
$t^4-t^2+1=0, t=e^{(2m+1)\pi i/6}$  where $m=0,2,3,5$
$$\implies f(x)=\sum_{m=\{0,2,3,5\}}A_me^{(2m+1)\pi ix/6}$$
Now if $f(x)=f(x+T)\forall x,$ we need  $$e^{(2m+1)\pi ix/6}=e^{(2m+1)\pi i(x+T)/6}$$
$\implies\dfrac{(2m+1)\pi T}6\equiv0\pmod{2\pi}$
$\iff(2m+1)T\equiv0\pmod{12}$
As $(2m+1,12)=1$ for $m=0,2,3,5,$ we need $$T\equiv0\pmod{12}$$