How many functions respect the equation $3f(f(x))-7f(x)+2x=0$

Question

How many functions $f:\mathbb Z\to \mathbb Z$ respect the equation $$3f(f(x))-7f(x)+2x=0$$

I the number of functions that respect the relation. I found $f(x)=2x$ and $f(x)=\frac {2x}6$. Is there another?

Answer 1

Consider $g(x)=f(x)-2x$. If $f:\mathbb{Z}\to\mathbb{Z}$ then $g:\mathbb{Z}\to\mathbb{Z}$. $$3f(f(x))=7f(x)-2x \Rightarrow 3g(f(x))=3f(f(x))-6f(x)=f(x)-2x=g(x)\Rightarrow g(f(x))=\frac{g(x)}{3}$$

Suppose that $g(x_1)\neq 0$ for some $x_1\in \mathbb{Z}$. Consider infinite sequence $(a_n)$, such that $a_i=|g(b_i)|$, $b_1=x_1$, $b_{i+1}=f(x_i)$. Then $a_{i+1}=\frac{a_i}{3}$ and $a_i\in\mathbb{Z}$. Sequence $(a_n)$ is decreasing, positive and integer. That is impossible. Contradiction.

Then $g(x)=0$ for all $x\in\mathbb{Z}$. Then $f(x)=2x$ is the only function $\mathbb{Z}\to\mathbb{Z}$ to satisfy the given equation.

Answer 2

Fix $x$ and let $a_n=f^n(x)$ (n-th iteration of $f$). Then we have $$3a_{n+1} -7a_n+2a_{n-1}=0$$ Solving this recurrence we get $$ a_n = a\cdot 2^n+b\cdot {1\over 3^n}$$ for some $a,b$. Since $a_0=x$ and $a_1 = f(x)$ we have $x=a+b$ and $f(x) = 2a +b/3$. Solving this on $a,b$ we get $$a={3f(x)-x\over 5}\;\;\;\;{\rm and} \;\;\;\;b = {6x-3f(x)\over 5}$$ Since $x,f(x)$ are integers we have $a=c/5$ and $b=d/5$ for some integers $c,d$. So we have $$5a_n - c\cdot 2^n=d\cdot {1\over 3^n}\implies 3^n\mid d \;\;\;\;\forall n\in \mathbb{N}$$ which is possible only iff $d=0$ and so $b=0$. Thus we have $f(x) =2x$.