## Covering systems with large moduli associated with reducible shifts of integer polynomials

### Volume 208 / 2023

#### Abstract

A variation of Turán’s polynomial conjecture is studied. Various connections between specific covering systems of congruences and reducible shifts of integer polynomials are established. These results are inspired by related work of A. Schinzel. As applications, it is shown that given an integer polynomial $f(x)$ with ${\rm deg}\,f \gt 0$, there is an integer $\lambda $ satisfying $\lvert \lambda \rvert \le 4\sqrt{{\rm deg}\,f}$ such that $x^{n}+f(x)+\lambda $ is irreducible over the rationals for infinitely many integers $n\ge 1$. Furthermore, if ${\rm deg}\,f \le 100$, then a desired $\lambda $ satisfying $\lvert \lambda \rvert \le 3$ exists.