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Let $f$ be a polynomial with complex coefficients and $n\in\mathbb N$ such that

  • $\deg(f)\leq n$
  • $f(0), f(1),...,f(n)\in\mathbb Z$

Prove or disprove each of the statements

  • $f(\mathbb Z)\subseteq\mathbb Z$ (in other words, for each integer $z\in\mathbb Z$ we also have $f(z)\in\mathbb Z$)
  • $f(\mathbb Z+i\mathbb Z)\subseteq\mathbb Z+i\mathbb Z$ (in other words, for each number $z=a+bi$ with $a,b\in\mathbb Z$ we also have $f(z)=c+di$ for some $c,d\in\mathbb Z$)

Source: I thought of it myself, but I'm probably not the first one to think of it.

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  • $\begingroup$ Welcome to PSE (Puzzling Stack Exchange)! $\endgroup$ Commented Jan 18 at 17:57

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Question 1:

If $f$ has degree at most $n$, then its $n$'th differences are constant. Since $f(0)$ through $f(n)$ are all integers, every entry in the difference triangle will also be an integer, including the $n$'th difference. Therefore, extending the top row in both directions will produce integers at every step.

Question 2:

The sequence $0,1,3$ is generated by $f(x) = \frac{x^2+x}{2}$, but $f(1+i) = \frac{1+3i}{2}$.

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