Question 136
A function $f: \mathbb{N} \to \mathbb{N}$, where $\mathbb{N}$ is the set of all positive integers, satisfies the following condition: for any positive integers $m$ and $n$ ($m \gt n$) the number $f(m) - f(n)$ is divisible by $m - n$.
Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n) = p(n)$ for all positive integers $n$?)
Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n) = p(n)$ for all positive integers $n$?)