# Prime polynomial

No polynomial $f (n)$ with integral coefficients, not a constant, can be prime for all $n$, or for all sufficiently large $n$.
We may assume that the leading coefficient in $f(n)$ is positive, so that $f(n)\rightarrow\infty$ when $n\rightarrow\infty$, and $f(n) > 1$ for $n > N$, say. If $x > N$ and $f(x)=a_0x^k+\ldots =y>1$ then $f(ry+x)=a_0\left(ry+x\right)^k+\ldots$ is divisible by $y$ for every integral $r$; and $f(ry+x)$ tends to infinity with $r$. Hence there are infinitely many composite values of $f(n)$.