Let $f(x)=a_0+a_1x+ \ldots +a_nx^n$ be a polynomial with integer
coefficients, where $a_n>0$ and $n \ge 1$. Prove that $f(x)$ is
composite for infinitely many integers $x$.
I can easily show that there are infinitely many composite numbers of the form $a_0+a_1x+ \ldots +a_nx^n$ if $a_0 \ge 2$, we just note that $f(x)$ is composite for every $x$ being a multiple of $a_0$. But I can't find a way to prove this in the case $a_0=1$.
Best Answer
Choose $m$ such that $f(m)\ne\pm1$, then choose any prime $p$ dividing $f(m)$, and think about $f(m+pk)$ for $k=1,2,\dots$.