1
$\begingroup$

I was solving a particular problem (not necessary here) when the following came to my mind.

Prove or disprove the statement:

If $P(x)$ be a polynomial such that $P(n) \in Z$ $ \forall n \in Z$ (Z denotes set of integers), then all coeffcients of $P(x)$ are integers.

Is this statement true? I haven't seen this proved ordisproved anywhere. Igot a feeling this may be false but no evidence.

  • 0
    For instance, $\frac{1}{30}n^5 - \frac{1}{30}n$ is always an integer. – 2017-01-06
  • 1
    More generally, fix any $k\ge 0$ and consider the function $n \mapsto {n \choose k}$. It's a polynomial that always takes integer values, but for $k>1$ the coefficients are not (all) integers. In a certain sense these are the only examples, as all other examples can be built by adding these together in various ways. – 2017-01-06

1 Answers 1

2

A simple counterexample?

Note that for a natural number $n$ the polynomial $$\binom {X}{n} =\frac {X (X-1)\cdots (X-n+1)}{n (n-1)\cdots 1}$$ takes integer values at all integers although it does not have integer coefficients.

Also see here. Hope it helps.

  • 0
    (except when $n=1$) – 2017-01-06