Functional equation $f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$

Question

Find all of the functions defined on the set of integers and receiving the integers value, satisfying the condition:

$$f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$$

for each pair of integers $(a,b)$.

Answer 1

Not cleaned. I just wrote it as I thought about it.

$A^3+B^3+C^3-3ABC=(A+B+C)(A+wB+w^2C)(A+w^2B+wC)$. So in our case

$$\left(f(a+b)-f(a)-f(b)\right)\left(f(a+b)-wf(a)-w^2f(b)\right)\left(f(a+b)-w^2f(a)-wf(b)\right)=0.$$

If for some $a,b$ we have $f(a+b)-wf(a)-w^2f(b)=0$ then $f(a+b)-wf(a)-w^2f(b)$ is divisible by $w^2+w+1$. Using long division we get $f(a+b)-wf(a)-w^2f(b)=(w^2+w+1)(-f(b))+(f(b)-f(a))w+f(a+b)-f(b)$. From where $f(b)=f(a)$ and $f(a+b)=-f(b)$.

So we are getting that, for each $a,b$, either:

$$f(a+b)=f(a)+f(b)$$

or $$f(a)=f(b)=-f(a+b).$$

We clearly get that $f(0)=0$.

If for some $x$, $f(x)=f(-x)$ then we get $f(x)=f(-x)=0$, otherwise we get $f(x)=-f(-x)$. If for some $x$, $f(x)=0$, then $f(nx)=0$.

If $f(1)=0$, then $f(a+1)=f(a)$. So, $f$ is constant equal to zero.

If $f(1)=r\neq0$ then $$f(a+1)=f(a)+r$$. Therefore $f(x)=rx$.