Given, $a,b,c\in\mathbb{R}$. Let $p(x)=ax^2+bx+c$. Find equivalent condition that always has an integer value ($p(x)$'s value) for $x\in\mathbb{Z}$.

Question

Problem : Given, $a,b,c\in\mathbb{R}$. Let $p(x)=ax^2+bx+c$. Then, find an equivalent condition that always has an integer value ($p(x)$'s value) for an arbitrary integer $x$.
[Problem is represented in Hungary Eotvos $1902$]

Answer 1

If you put $x = -1, 0, 1$, then you get $c\in \mathbb{Z}$, $a+b, a-b\in \mathbb{Z}$, which implies $a = \frac{m}{2}, b = \frac{n}{2}$ for some $m, n\in \mathbb{Z}$ with $m\equiv n(\mathrm{mod}\,2)$. Now try to show the converse - if $a, b, c$ are of this form, then $f(x)\in \mathbb{Z}$ for any $x\in \mathbb{Z}$.