PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On the nearest irreducible lacunary neighbour to an integer polynomial

Volume 162 / 2020

Pradipto Banerjee, Ranjan Bera Colloquium Mathematicum 162 (2020), 121-134 MSC: Primary 11C08; Secondary 11R09, 12E05. DOI: 10.4064/cm7978-8-2019 Published online: 15 April 2020

Abstract

There is an absolute constant $D_{0} \gt 0$ such that if $f(x)$ is an integer polynomial, then there is an integer $\lambda $ with $|\lambda | \le D_{0}$ such that $x^{n}+f(x)+\lambda $ is irreducible over the rationals for infinitely many integers $n\ge 1$. Furthermore, if $\deg f \le 25$, then there is a $\lambda $ with $\lambda \in \{-2,-1,0,1,2,3\}$ such that $x^{n}+f(x)+\lambda $ is irreducible over the rationals for infinitely many integers $n\ge 1$. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pál Turán.

Authors

  • Pradipto BanerjeeIndian Institute of Technology Hyderabad
    Kandi, Sangareddy
    Telangana 502285, India
    e-mail
  • Ranjan BeraIndian Institute of Technology Hyderabad
    Kandi, Sangareddy
    Telangana 502285, India
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image