Value of m for which the function will give integers as an output.

Question

$F(m)=(2m^3+2m)/(m^2+1)$ and $g(m)=(m^4+1)/(m^2+1)$ What are the values of $m$ other than $1$ for which solution of both function will be integers. Please tell if there is any formula to find so or any technique?

Answer 1

Whenever $m^2+1 \ne 0$, you find that $F(m)=2m$ and $g(m)=m^2-1+2/(m^2+1)$, so $F(m)$ will be an integer for all integers $m$.

Assumig that only integers are allowed for $m$, $g(m)$ will be an integer just for $m \in \{-1,0,1\}$.

Answer 2

Well, notice that $\forall x\in\mathbb{Z}$:

$$\frac{2x^3+2x}{x^2+1}=2x\tag1$$

So:

$$2x\in\mathbb{Z}\tag2$$