A necessary and sufficient on the co-efficients of a quadratic to give an integer

Question

Le $f(n) := an^2 + bn + c$ for all integers $n$, where $a$, $b$, $c$ are rational. What are the necessary and sufficient conditions on $a$, $b$, and $c$ such that $f(n)$ be an integer for all $n$?

Answer 1

There is an answer for polynomials of arbitrary degree $d$. Write it in the form $p(x)=\sum_{i=0}^d c_i \binom{x}{i}$, where $\binom{x}{i}=x(x-1)\dots(x-i+1)/i!$. Then $p(n)$ is integer for every integer $n$ iff $c_i$'s are integers. [to see it: if $c_i$'s are integers then $\sum_{i=0}^d c_i \binom{n}{i}$ is certainly an integer; to see the other implication, look at the polynomials $\Delta p(x):=p(x+1)-p(x)$, $\Delta^2 p(x)$, etc, and notice $p(0)=c_0$, $\Delta p(0)=c_1$, $\Delta^2 p(0)=c_2$ etc].

For quadratic polynomials: $A\binom{x}{2}+B\binom{x}{1}+C=Ax^2/2+(B-A/2)x+C=ax^2+bx+c$; $A,B,C$ are integers iff $c$ is an integer and if both $a,b$ are integers or they are both (odd integer)/$2$.

Answer 2

We are told that $an^2+bn+c$ is an integer forevery integer value of $n$. Set $n=0$. This tells us that $c$ must be an integer.

Since $c$ is an integer, $an^2+bn+c$ is an integer for all $n$ if and only if $an^2+bn$ is an integer for all $n$.

Suppose that $an^2+bn$ is an integer for all $n$. Then in particular it is an integer for $n=1$ and for $n=-1$. Thus $a+b$ is an integer, say $k$, and $a-b$ is an integer, say $l$.

From $a+b=k$, $a-b=l$ we get $a=\frac{k+l}{2}$ and $b=\frac{k-l}{2}$. There are two possibilities (i) $k$ and $l$ have the same parity, that is they are both even or both odd and (ii) $k$ and $l$ have opposite parities.

In Case (i), the numbers $a$ and $b$ are integers.

In Case (ii), the numbers $a$ and $b$ are each half of an odd integer.

Finally, we check that (i) if $a$ and $b$ are integers, then $an^2 +bn$ is an integer for all integers $n$. This is obvious.

We also check that (ii) if $a$ and $b$ are each half of an odd integer, then $an^2+bn$ is an integer for all $n$.

So $a=\frac{c}{2}$ and $b=\frac{d}{2}$ where $c$ and $d$ are odd integers. We have then $an^2+bn=\frac{1}{2}n(cn+d)$. If $n$ is even, then certainly $\frac{1}{2}n(cn+d)$ is an integer. And if $n$ is odd, then $cn+d$ is even, so again $\frac{1}{2}n(cn+d)$ is an integer.

Conclusion: we have $an^2+bn+c$ is an integer for all integers $n$ if and only if $c$ is an integer and $a$ and $b$ are each integers, or each half of an odd integer.