Covering systems with large moduli associated with reducible shifts of integer polynomials
Tom 208 / 2023
Acta Arithmetica 208 (2023), 83-100
MSC: Primary 11R09; Secondary 11C08, 11B75.
DOI: 10.4064/aa220518-18-3
Opublikowany online: 17 April 2023
Streszczenie
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.
