A question on polynomials with integral coefficients.
By Sarah Rodriguez •
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.
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$\begingroup$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.
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