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

Volume 208 / 2023

Pradipto Banerjee Acta Arithmetica 208 (2023), 83-100 MSC: Primary 11R09; Secondary 11C08, 11B75. DOI: 10.4064/aa220518-18-3 Published online: 17 April 2023


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.


  • Pradipto BanerjeeIndian Institute of Technology Hyderabad
    Kandi, Sangareddy
    Telangana 502285, India

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