$\pi$ and the irreducibility of polynomials that take a prime value
Nicolae Bonciocat
Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Abstract:
Given an integral polynomial $f$, we show how $\pi$ is related to lower bounds on the positive integer values $m$ for which the primality of $f(m)$ guarantees the irreducibility of $f$. Our study relies on finding suitable zero-free angular sectors and zero-free lens-shaped regions for $f$ and its reciprocal. The results are based on joint work with C.M. Bonciocat.