Many Multiples

Algebra Level 4

Let $f(x)$ be a non-constant polynomial with integer coefficients.

Is it true that for every integer $n$ that is not a root of $f(x)$ , there exist infinitely many integers $m$ such that $|f(m)|$ is a distinct integral multiple of $|f(n)|$ ?

No Yes

1 solution

Brandon Monsen
Aug 7, 2017

Let $f(n)=\alpha$ for some integer $n$ . Then $f(n+k\alpha)$ is a multiple of $\alpha$ for all integers $k$ .

We know that this must hit infinitely many distinct multiples of $\alpha$ since if it didn't, then that would imply that one multiple of $\alpha$ , call it $c$ , was crossed infinitely many times, which is analogous to saying that there are infinitely many roots to the polynomial $f(x)=c$ . This is clearly impossible since $f(x)$ is of finite degree.

Many Multiples

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