Polynomials with no small prime values

Abstract:

Let $f(x)$ be a polynomial with integer coefficients, and let \[ D(f) = {\text {g}}{\text {.c}}{\text {.d}}\{ f(x):x \in {\mathbf {Z}}\}.\] It was conjectured by Bouniakowsky in 1857 that if $f(x)$ is nonconstant and irreducible over ${\mathbf {Z}}$, then $|f(x)|/D(f)$ is prime for infinitely many integers $x$. It is shown that there exist irreducible polynomials $f(x)$ with $D(f) = 1$ such that the smallest integer $x$ for which $|f(x)|$ is prime is large as a function of the degree of $f$ and the size of the coefficients of $f$.