Question 57
Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that $f(0)=2$ and for all integers $x$,
$$
f(x+1)+f(x-1)=f(x) f(1) .
$$
Prove that for all integers $x$ and $y$,
$$
f(x+y)+f(x-y)=f(x) f(y) .
$$
$$
f(x+1)+f(x-1)=f(x) f(1) .
$$
Prove that for all integers $x$ and $y$,
$$
f(x+y)+f(x-y)=f(x) f(y) .
$$