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Let $f,g \in \mathbb{Z}[x],\deg(f),\deg(g)>1$ and $f$ is monic.

Assume $f$ and $g$ are coprime.

For integer $a$ is it possible $g(a) \mid f(a)$ many times?

Is it possible unbounded number of times for fixed degree?

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If the polynomials $f,g$ are coprime, then there are integer polynomials $f_1,g_1$ such that $d=ff_1+gg_1$ is a nonzero constant (it's true if we allow $f_1,g_1$ to have rational coefficients, then we just clear denominators). If $g(a)\mid f(a)$, then $g(a)\mid d$, from where it's clear that for fixed pair $f,g$ there are only finitely many $a$ like you request.

If we only bound the degree, we can have arbitrarily many divisibilities even in the simplest case $\deg f=\deg g=2$. Namely take $f(x)=x^2+(n^2)!,g(x)=x^2$ for arbitrary $n$. Then for $1\leq a\leq n$, $g(a)=a^2\mid a^2+(n^2)!=f(a)$.

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  • $\begingroup$ Is it still easy if we require $g$ to be squarefree? $\endgroup$
    – joro
    Commented Oct 28, 2017 at 14:45
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    $\begingroup$ @joro Yes, make it $g(x)=x^2+1,f(x)=x^2+1+(n^2+1)!$. $\endgroup$
    – Wojowu
    Commented Oct 28, 2017 at 14:47

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