Let $f(x)\in\mathbb{Q}[x]$ be a polynomial of degree $n$ such that $f(\mathbb{Z})\subseteq \mathbb{Z}$.
I want to show that $f$ has the following form $$f(x)=\sum_{j=0}^{j=n} a_{j}\binom{x}{j}$$ with $a_{j}\in \mathbb{Z}$
Attempt:
Base case $n=0$, is clear
Induction Hypothesis: Assume the result for degree$=n-1$
Consider the polynomial $\Delta f(x)=f(x+1)-f(x)$. One can observe that it has degree $n-1$, so $\Delta f$ has the above mentioned form.
Can I deduce something from here ?
Any other approach ?
Kindly correct the tags if necessary, I am studying Hilbert Polynomial and Hilbert Series.
If I am correct, The above problem characterize all numerical polynomial.